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This paper deals with a numerical approximation of the elasticity vibration problem based on a potentials decomposition. Decomposing the displacements field into potentials is a well-known tool in elastodynamics that takes advantage of the decoupling of pressure waves and shear waves inside a homogeneous isotropic media. In the spectral problem on a bounded domain, this decomposition decouples the elasticity equations into two Laplacian-like equations that only interact at the boundary. We show that spurious eigenvalues appear when Lagrangian finite elements are used to discretize the problem. Then, we propose an alternative weak formulation which avoids this drawback. A finite element discretization of this weak formulation based again on Lagrangian finite elements is proposed and tested by means of some numerical experiments, which show convergence and absence of spurious modes. [ABSTRACT FROM AUTHOR]

This paper develops a fully discrete two-grid mixed finite element scheme of a phase field model for the two immiscible and incompressible flows. In the model, the Navier-Stokes equations and Allen-Cahn equation are coupled by a phase function in which an extra phase induced stress term is added in the Navier-Stokes equations and a fluid induced transport term is added in the Allen-Cahn equation. We use the two-grid method to overcome the computational difficulties arising from the strong nonlinear coupling system. The stability and convergence of the two-grid method are obtained under the proper assumption of time discretization size k , space discretization size h and interface thickness ε. The analysis shows that the two-grid methods can get the same convergence as the classical one-grid methods provided that we choose proper coarse and fine mesh sizes. However, the two-grid methods can save much computational time compared with the one-grid methods. Some numerical experiments are provided to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new dynamical threshold based Perona–Malik (DTPM) model for image denoising. As one of the most famous anisotropic diffusion equations, the PM model has been widely used in noise removal, image segmentation, edge detection and image enhancement. However, the major disadvantages of the traditional PM model are the high difficulty of choosing the best threshold and the tendency of impairing details, so that the denoising is either excessive or insufficient in the whole process. These defects are more obviously when dealing with the image that polluted by noise from multiple distributions. By designing a dynamical threshold function in the edge indicator, we establish a new PM model that can change the diffusion mode and strength adaptively according to the image features. The existence and uniqueness of the weak solution are proven. After theoretical analysis, we give an experimental approach to show the efficiency of this kind of model, especially to the image containing multi–noise and multi–details. [ABSTRACT FROM AUTHOR]

In this paper, we consider the backward problem for an inhomogeneous time-fractional diffusion-wave equation in an axis-symmetric cylinder. This problem is ill-posed. In order to solve it stably, we give an iterative regularization method. The Hölder error estimates are obtained under the a-priori and a-posteriori parameter choice rules. Numerical examples are presented to verify the effectiveness of our method. [ABSTRACT FROM AUTHOR]

In this paper, we study an unconditionally stable and decoupled identical-relation-preserving scheme for solving the incompressible Navier-Stokes equations with variable density. By introducing power- and exponential-type scalar auxiliary variables to define the system's energy and to balance the incompressible condition's influence respectively, we first convert the Navier-Stokes equations with variable density into an equivalent form. Then, with the help of the equations of scalar auxiliary variables, we construct a linear decoupled fully discrete finite element scheme, which can be implemented efficiently. Moreover, the mass, momentum and energy identical relations which are analogous to that for the continuous equations are proved uniformly regardless of the discretization parameters. Both lower- and upper-bound of the numerical density measured in L 2 -norm are deduced, too. Finally, we report some numerical examples to verify the correctness and efficiency of the proposed scheme, including an extension of a coupled nonlinear model with temperature. [ABSTRACT FROM AUTHOR]

In this paper an implicit peridynamic (PD) framework is devised for the solution of quasi-static problems involving brittle and quasi-brittle fracture. A common way to deal with such problems, in explicit frameworks, is to incorporate artificial damping into the PD equation of motion. This may contribute to non-physical behavior of the system. Also, implicit frameworks, based on the Newton-Raphson method, fail to trace the snap-back behavior. To this end, we develop an implicit peridynamic framework to deal with quasi-static problems involving snap-back/snap-through instabilities. This is achieved through two different approaches. The first approach is based on the global arc-length method applicable to problems with zero displacement constraints. The second one is based on a displacement-controlled arc-length method, suitable for problems with non-zero prescribed displacements. The framework is developed for linear elastic as well as softening (degrading) material damage models, respectively appropriate for brittle and quasi-brittle fracture modeling. The robust performance of the proposed framework, within various loading scenarios and damage models, is demonstrated. To boost the numerical performance of the proposed PD framework, a hybrid integration scheme is employed. We demonstrate the advantages of this scheme over the conventional (standard) one in terms of efficiency for quasi-static problems. [ABSTRACT FROM AUTHOR]

In this paper we propose a solution strategy for the Cahn-Larché equations, which is a model for linearized elasticity in a medium with two elastic phases that evolve subject to a Ginzburg-Landau type energy functional. The system can be seen as a combination of the Cahn-Hilliard regularized interface equation and linearized elasticity, and is non-linearly coupled, has a fourth order term that comes from the Cahn-Hilliard subsystem, and is non-convex and nonlinear in both the phase-field and displacement variables. We propose a novel semi-implicit discretization in time that uses a standard convex-concave splitting method of the nonlinear double-well potential, as well as special treatment to the elastic energy. We show that the resulting discrete system is equivalent to a convex minimization problem, and propose and prove the convergence of alternating minimization applied to it. Finally, we present numerical experiments that show the robustness and effectiveness of both alternating minimization and the monolithic Newton method applied to the newly proposed discrete system of equations. We compare it to a system of equations that has been discretized with a standard convex-concave splitting of the double-well potential, and implicit evaluations of the elasticity contributions and show that the newly proposed discrete system is better conditioned for linearization techniques. [ABSTRACT FROM AUTHOR]

In this paper, an efficient dimension splitting method is applied to the two-dimensional (2D) and three-dimensional (3D) Cahn-Hilliard type equations with significant applications in physical, biological and computer science. The proposed method can greatly reduce the storage requirements and computational complexity in two and three dimensions, and preserve the high precision and mass conservation. The dimension splitting method is based on auxiliary variables for spatial derivative terms and the operator splitting approach. It converts the 2D or 3D problem into a series of one-dimensional (1D) problems which can be solved by using multi-thread. The stability and accuracy of the proposed method are improved by a local stabilizing approach and extrapolation. The analyses of discrete energy and discrete mass conservation property are shown. Numerical examples confirm the advantages of proposed method. And a variety of simulations, such as the phase separations, curvature driven flows, diblock copolymers, tumor growth and volume restoration, are performed with respect to practical applications. • Dimension splitting method for Cahn-Hilliard type equations. • The global stability is achieved by the local stability of solving each sub-problem. • Fourth-order mass conservative scheme based on the compact differencing and extrapolation. • A large number of numerical examples for practical applications. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a new mixed method proposed by Rafetseder and Zulehner for Kirchhoff plates and apply it to fourth order eigenvalue problems. Using two auxiliary variables this new mixed method makes it possible to require only H 1 regularity for the displacement and the auxiliary variables, without the demand of a convex domain. We provide a direct comparison, specifically in view of convergence orders, to the C 0 -IPG method and Ciarlet-Raviart's mixed method of vibration problems with the boundary conditions of the clamped plate and the simply supported plate. The numerical experiments are done with the open-source finite element library deal.II and include the implementation of the coupling of finite elements with elements on the boundary to incorporate non-trivial boundary conditions. [ABSTRACT FROM AUTHOR]

In this paper, the pressure correction finite element method is proposed for the 2D/3D time-dependent thermomicropolar fluid equations. The first-order and second-order backward difference formulas (BDF) are adopted to approximate the time derivative term, stability analysis and error estimation of the first-order semi-discrete scheme are proved. Finally, some numerical examples are given to show the effectiveness and reliability of the proposed method, which can be used to simulate the problem with high Rayleigh number. [ABSTRACT FROM AUTHOR]

The problem of multiplicative noise removal has attracted much attention in recent years. Unlike additive noise, multiplicative noise destroys almost all information of an image; therefore, it is more difficult to remove them from corrupted images. In this paper, we propose a denoising model of fractional-order diffusion coupled with integer-order diffusion for multiplicative gamma noise, which takes advantage of both texture-preserving property of fractional-order diffusion and edge-preserving property of integer-order diffusion. We explore the mutual mechanism of two diffusion equations in the diffusion process, i.e., mutual transfer of texture and edge information respectively from two filtering images, and mutual regularization on the diffusion coefficients in both equations. We design an alternating numerical scheme based on semi-implicit finite difference and discrete Fourier transform for solving the coupled diffusion system. The proposed model is tested on some commonly-used images and is compared with six state-of-the-art models, qualitatively and quantitatively. Experimental results show the superiority of the proposed model in reducing multiplicative gamma noise while preserving textures and edges. [ABSTRACT FROM AUTHOR]

Even when the wave equation is a classical problem to solve in courses about hyperbolic partial differential equations, the numerical solution for this equation in irregular regions has been little studied. This is because of all the problems that could arise when solving the second-order derivative in time. This paper presents a generalized finite difference scheme applied to the numerical solution of the wave equation, solved on clouds of points for highly irregular domains. The implementation over clouds of points allows this scheme to compute satisfactory results, with minor quadratic errors, over highly irregular regions, as discussed in the present manuscript. Also, the numerical results show the accuracy obtained with a straightforward and low-cost implementation. [ABSTRACT FROM AUTHOR]

The implementation methods of finite element analysis (FEA) have remained essentially unchanged since the inception of FEA in the 1960s. Alterations of any of the input or design parameters to the FEA model can potentially nullify the previous results and subsequent additional simulations will be required. This is particularly relevant for situations that require active monitoring where telemetry is to be passed to remote systems capable of carrying out FEA computations. In this paper, we train an artificial neural network that was originally developed for image processing to emulate FEA. Conventionally generated FEA results are transformed into image pairs where the load, material and geometric properties are assigned different colour channels. These images are used to train a conditional Generative Adversarial Network (cGAN). The subsequent "trained" cGAN can generate predictions for arbitrary inputs which correspond to the domain of input on which the developed cGAN was trained. Three numerical experiments were conducted resulting in three separate cGANs trained to infer (a) deflections from forces, (b) stresses from deflections and (c) stresses from forces. After a moderate training regime of 200 epochs each, the outputs of the trained networks are shown to be in reasonable agreement to the ground truth with mean errors in the range of 5-10%. The contribution of this work lies in transforming FEA results into images which enables the usage of cGANs to solve a computational mechanics problem. The implementation herein allows for near real-time computations which highlights the potential of the proposed methodology in applications where simulation results are required in a timely manner such as predictive control, interactive virtual environment, etc. All the codes used in this research will be openly available at Qatar University's Institutional Repository1; the data used in this work will be available upon request from the corresponding author. [ABSTRACT FROM AUTHOR]

In this paper, the regularity of the solution for a multi-term time-fractional fourth-order equation is analyzed, and the derived result shows that the solution behaves a weak singularity at initial time t = 0. By introducing an intermediate variable p = − Δ u , we obtain a system of two second-order differential equations, which is discretised by using the L1 scheme on graded meshes in time and the mixed finite element method (MFEM) in space. It is shown that the optimal pointwise-in-time error estimate is derived for the proposed scheme. Moreover, the fast L1-MFEM is constructed to improve the computational efficiency. Finally, numerical experiments are provided to further verify our theoretical convergent result and the computational efficiency of the proposed two schemes. [ABSTRACT FROM AUTHOR]

In this paper, a Galerkin finite element method is designed and analyzed to simulate the nonlinear Korteweg-de Vries-Rosenau-regularized long-wave (KdV-RRLW) model. We establish the existence and uniqueness results in H 0 2 (Ω) Sobolev space by applying the Banach–Alaoglu theorem. Using appropriate projection, we derive the error estimates of a semidiscrete scheme for the finite element solution of the KdV-RRLW model. Furthermore, a second-order Crank-Nicolson scheme is employed for the temporal discretization and obtain the optimal order of convergence in the maximum norm. Finally, several numerical examples are provided to visualize the nature of the wave phenomena in one and two dimensional spaces and demonstrate the robustness of the developed finite element algorithm. [ABSTRACT FROM AUTHOR]

This paper focus on construction of high-order energy-preserving difference scheme for the fourth-order nonlinear strain wave equation with an energy conservation law. This target model is firstly transformed into an equivalent system by using the method of trigonometric scalar auxiliary variables. The resulting equivalent system possess a modified energy conservation law, and a fourth-order difference scheme with analogously discrete energy conservation law is developed based on the resulting equivalent system. The boundedness and convergence of the numerical solutions in the maximum norm are shown. The effectiveness of the difference scheme is verified by several numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we use direct meshless local Petrov-Galerkin (DMLPG) methods for solving the variable-order time-fractional mobile/immobile advection-diffusion equation in two dimensions. The basis of the DMLPG methods is on the generalized moving least-square (GMLS) approximation and local weak form of the equation. Since the GMLS approximation uses basic polynomials as shape functions, there is no need to construct complex shape functions in the moving least-square (MLS) approximation. In addition, the calculation of numerical integrals in the local weak form of the equation is simple. We also give proof of the convergence and stability of the time-discrete scheme. The analytical results show that the semi-discrete time scheme is unconditionally stable and the convergence order is O (Δ t 2). Moreover, to show the accuracy and efficiency of the methods, we use the regular and irregular domains with distributions of uniform and scattered nodes in numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we propose an unconditionally stable numerical technique for a multi-dimensional Black-Scholes equation to price an option with high accuracy. The proposed scheme uses the operator splitting method to reduce the multi-dimensional partial differential equation to a set of one-dimensional sub-problems. The computational domain is discretized with a uniform space and time step size to approximate the option values and asset correlation-related terms by piecewise quadratic polynomials. In order to obtain the numerical solutions of the sub-problems, we analytically integrate the polynomials to estimate the expectation of the Feynman-Kac formula. We compare our method with the implicit operator splitting method (OSM), a widely used finite difference method in the industry. Numerical experiments show that the proposed method outperforms OSM in terms of convergence in space and time directions. We also provide analysis to guarantee the unconditional stability of our method by exploiting the Feynman-Kac recursively. [ABSTRACT FROM AUTHOR]

In the present paper, a new strategy of multigrid method is introduced to accelerate the convergence speed of numerical simulations via lattice Boltzmann method. Based on the popular V-cycle multigrid algorithm, a simplified multigrid algorithm is presented and validated through the simulations of the classic lid-driven cavity flow for steady states. The novelty of the present algorithm resides in the construction of the information transferring process, in which, for a full cycle, the numerical simulation starts on the coarse mesh, transferring to medium mesh and then streams to the fine mesh through a prolongation operator. Afterwards, instead of using a restriction operator, the fluid information jumps back directly from the fine mesh to the coarse mesh via an assignment operator. The accuracy and efficiency of the simplified new algorithm are validated by comparing the results obtained when employing the classic V-cycle multigrid algorithm and the traditional lattice Boltzmann method with uniform Cartesian grid. [ABSTRACT FROM AUTHOR]

We design a virtual element method for the numerical treatment of the two-dimensional parabolic variational inequality problem on unstructured polygonal meshes. Due to the expected low regularity of the exact solution, the virtual element method is based on the lowest-order virtual element space that contains the subspace of the linear polynomials defined on each element. The connection between the nonnegativity of the virtual element functions and the nonnegativity of the degrees of freedom, i.e., the values at the mesh vertices, is established by applying the Maximum and Minimum Principle Theorem. The mass matrix is computed through an approximate L 2 polynomial projection, whose properties are carefully investigated in the paper. We prove the well-posedness of the resulting scheme in two different ways that reveal the contractive nature of the VEM and its connection with the minimization of quadratic functionals. The convergence analysis requires the existence of a nonnegative quasi-interpolation operator, whose construction is also discussed in the paper. The variational crime introduced by the virtual element setting produces five error terms that we control by estimating a suitable upper bound. Numerical experiments confirm the theoretical convergence rate for the refinement in space and time on three different mesh families including distorted squares, nonconvex elements, and Voronoi tesselations. [ABSTRACT FROM AUTHOR]

• A new coupled framework based on the ALE-DUGKS is introduced to study the nonlinear SFD problem. • Linear and tilting motions of a rigid micro-beam are studied under forced and free oscillations. • The cause of the nonlinear damping phenomenon is investigated. In this paper, the nonlinear squeeze-film damping (SFD) involving rarefied gas effect in the micro-electro-mechanical systems (MEMS) is investigated. Considering the motion of structures (beam, cantilever, and membrane) in MEMS, the dynamic response of the structure is affected greatly by the SFD. In the traditional model, a viscous damping assumption that the damping force is linear with the moving velocity is used. As the nonlinear damping phenomenon is observed for a micro-structure oscillating at a high velocity, this assumption does not hold and will cause error results for predicting the response of the micro-structure. Meanwhile, due to the small size of the device and the low pressure of the encapsulation, the gas in MEMS is usually rarefied gas. Therefore, to correctly predict the damping force, the rarefied gas effect must be considered. To study the nonlinear SFD problem involving the rarefied gas effect, a kinetic method, i.e., discrete unified gas kinetic scheme (DUGKS), is introduced in this paper. Also, based on the DUGKS, two solving methods, i.e., a traditional decoupled method (Eulerian scheme) and a coupled framework (arbitrary Lagrangian-Eulerian scheme), are adopted. With these two methods, two basic motion forms, i.e., linear (perpendicular) and tilting motions of a rigid micro-beam, are studied under forced and free oscillations. For a forced oscillation, the nonlinear SFD phenomenon is investigated. For a free oscillation, in the resonance regime, some numerical results at different maximum oscillating velocities are presented and discussed. Besides, the influence of oscillation frequency on the damping force or torque is also studied, and the cause of the nonlinear damping phenomenon is investigated. [ABSTRACT FROM AUTHOR]

To solve general sparse linear systems, this paper presents a domain decomposition based parallel preconditioner. Vertex-based partitioning is utilized to reorder the original coefficient matrix, resulting in a s × s block structure. Here, s is the number of subdomains used in the partition. Variables corresponding to the interface nodes are obtained by solving a linear system with coefficient matrix being the Schur complement S of the reordered matrix. Combining local low-rank correction approximation with a global low-rank correction technique to approximate the inverse of S , the method presented in this paper is different with previous Schur complement based preconditioners. The global low-rank correction terms are obtained by using the information comes from the local low-rank correction terms. In addition, variables corresponding to the interior nodes are computed by solving s small linear systems in parallel. Some numerical tests are presented to show the efficiency and robustness of the proposed preconditioner. [ABSTRACT FROM AUTHOR]

Images taken and stored digitally are often degraded by noise, so that the perceived image quality is significantly decreased in the presence of noise and human gaze behavior is affected by this noise as well. In recent years, image denoising has been modeled as partial differential equations (PDEs), which can be said to eliminate noise as well as preserve image detail is one of their main advantages. Numerical simulation of these PDEs, which are often nonlinear, high-order and high-dimensional, is one of the most challenging topics in this field. In this paper, one of the most powerful meshless numerical approaches, the operator splitting radial basis function (RBF) collocation method, is introduced to solve two modified versions of the Perona-Malik (PM) equation. The method splits each problem into several equations each of which is solved in one direction so as to make the major problem solvable in a smaller size. Furthermore, being in one direction makes it possible to solve the equations without any need for domain decomposition and its complexities. An unconditionally stable method (Crank-Nicolson) is used for discretizing time in this paper. Using meshless approaches in the image processing problems makes sense since the structure of digital images consist of pixels that corresponds to the distributed nodes of the meshless method so the complex time consuming domain construction of the finite element methods is avoided to get higher speeds in obtaining the desired denoised results. Some images with different noise levels are denoised using the presented method in the section of numerical results to show the desirable performance of the method. Also, the quality of the denoised images are concluded in tables which can be used to compare the obtained results. [ABSTRACT FROM AUTHOR]

This paper focuses on the weakly singular integrals in thin structures of 3D boundary element method. In the pioneering woks for the weakly singular integrals, the singularities are eliminated through introducing transformations whose Jacobian at the singular point is zero. When using these methods for weakly singular boundary integrals on elements with a large-angle and large side length ratio, near singularity will appear. In this paper, a dual transformation method is proposed to accurately calculate weakly singular integrals in which the near singularity is also considered. The numerical implementation is divided into several steps. Firstly, the (α , β) coordinate transformation method is used to eliminate the weak singularity by introducing the α factors in the transformation Jacobian. Then extract the integral form with near singularities about β. Finally, a corresponding distance transformation is constructed to eliminate its near singularity. The results show that the proposed method can accurately integrate for arbitrary element shape and source point position. The dual transformation method may be a good choice to evaluate weakly singular integral for thin structural problem. [ABSTRACT FROM AUTHOR]

• Proposing a self-adaptive algorithm (ANE) for designing a nearly optimal two-layer NN for solving PDEs. • A method of continuation by the ANE method for providing a good initialization in training the neural network. • Analyzing effect of numerical integration. • Introducing a posteriori error estimators of recovery type for the ANE method. • Demonstrating superior performance of the ANE method for problems with interface singularities and sharp interior layers. In the companion paper [1] , we introduce adaptive network enhancement (ANE) method for the best least-squares approximation to a target function by using two-layer ReLU neural networks (NNs). In this paper, we apply the ANE method for solving self-adjoint second-order elliptic partial differential equations (PDEs). The underlying PDE is discretized by the Ritz method using a two-layer spline neural network based on either the primal or dual formulations that minimize the respective energy or complimentary functionals. Essential boundary conditions are imposed weakly through the functionals with proper norms. It is proved that the Ritz approximation is the best approximation in the energy norm; moreover, effect of numerical integration for the Ritz approximation is analyzed as well. Two estimators for adaptive neuron enhancement method are introduced, one is the so-called recovery estimator and the other is the least-squares estimator. Finally, numerical results for diffusion problems with either corner or intersecting interface singularities are presented. [ABSTRACT FROM AUTHOR]

A convergence theory for the hp -FEM applied to a variety of constant-coefficient Helmholtz problems was pioneered in the papers [35] , [36] , [15] , [34]. This theory shows that, if the solution operator is bounded polynomially in the wavenumber k , then the Galerkin method is quasioptimal provided that h k / p ≤ C 1 and p ≥ C 2 log ⁡ k , where C 1 is sufficiently small, C 2 is sufficiently large, and both are independent of k , h , and p. The significance of this result is that if h k / p = C 1 and p = C 2 log ⁡ k , then quasioptimality is achieved with the total number of degrees of freedom proportional to k d ; i.e., the hp -FEM does not suffer from the pollution effect. This paper proves the analogous quasioptimality result for the heterogeneous (i.e. variable-coefficient) Helmholtz equation, posed in R d , d = 2 , 3 , with the Sommerfeld radiation condition at infinity, and C ∞ coefficients. We also prove a bound on the relative error of the Galerkin solution in the particular case of the plane-wave scattering problem. These are the first ever results on the wavenumber-explicit convergence of the hp -FEM for the Helmholtz equation with variable coefficients. [ABSTRACT FROM AUTHOR]

In this paper, we study the nonlocal nonlinear Schrödinger equation (NNLSE), which is an important extension of the nonlinear Schrödinger equation (NLSE). Since the nonlinearity of NNLSE involves a convolution, numerical approximation of its solution is very expensive. To achieve an efficient numerical method for solving the NNLSE, according to the property of convolution, we firstly present a partial differential equation (PDE) method to transfer the NNLSE from an integro-differential equation to an equivalent or approximate system of PDEs, and the numerical solution of the NNLSE can then be obtained by solving these PDEs. Based on the recently developed finite integration method (FIM), we derive an improved method (IFIM) in this paper. The novelty of this IFIM is that the quadrature method is only used once instead of multiple times for multi-layer integral, so it consumes less running time and provides higher accuracy. In addition, we adapt a second-order operator-splitting method (OS2) at each time-step to ensure a convergent solution for long-time integration. Several numerical experiments are given to verify the efficiency and accuracy of the proposed IFIM for solving the NNLSE. [ABSTRACT FROM AUTHOR]

This paper develops and investigates a new method for the application of Dirichlet boundary conditions for computational models defined by point clouds. Point cloud models often stem from laser or structured-light scanners which are used to scan existing mechanical structures for which CAD models either do not exist or from which the artifact under investigation deviates in shape or topology. Instead of reconstructing a CAD model from point clouds via surface reconstruction and a subsequent boundary conforming mesh generation, a direct analysis without pre-processing is possible using embedded domain finite element methods. These methods use non-boundary conforming meshes which calls for a weak enforcement of Dirichlet boundary conditions. For point cloud based models, Dirichlet boundary conditions are usually imposed using a diffuse interface approach. This leads to a significant computational overhead due to the necessary computation of domain integrals. Additionally, undesired side effects on the gradients of the solution arise which can only be controlled to some extent. This paper develops a new sharp interface approach for point cloud based models which avoids both issues. The computation of domain integrals is circumvented by an implicit approximation of corresponding Voronoi diagrams of higher order and the resulting sharp approximation avoids the side effects of diffuse approaches. Benchmark examples from the graphics as well as the computational mechanics community are used to verify the algorithm. All algorithms are implemented in the FCMLab framework and provided at https://gitlab.lrz.de/cie%5fsam%5fpublic/fcmlab/. Further, we discuss challenges and limitations of point cloud based analysis w.r.t. application of Dirichlet boundary conditions. [ABSTRACT FROM AUTHOR]

In [13] , we derived stress intensity factors (SIF) extraction formulas of a biharmonic equation Δ 2 u = f in a cracked domain whose crack faces have clamped boundary conditions. In this paper, we extend our investigation to the derivation of the SIF extraction formulas of the biharmonic equation in non-convex polygonal domains containing cracks or reentrant corners when various BC such as clamped (CC), simply supported (SS), free (FF), or mixed conditions (CS, FC, SF) are imposed on the boundary. We prove that the SIF is expressed as the integral of f Ψ s k ⁎ − u Δ 2 Ψ s k ⁎ for a cut-off function Ψ and a dual singular function s k ⁎ on a small neighborhood of the singularity. The dual singular function is determined in this paper. For a numerical approximation of u , we proposed an iteration method as well as a direct finite element method. For a finite element solution of Δ 2 u = f , we have to use C 1 -continuous basis functions. For continuous differentiable basis functions, we use either C 1 -continuous B-spline basis functions or the conventional Hermite basis function. Because of several advantages of B-spline functions in imposing complex boundary conditions, we choose B-spline basis functions for the finite element approximation of u. Moreover, in the direct numerical method, we propose to use implicitly enriched basis functions that resemble the singularities. [ABSTRACT FROM AUTHOR]

In this paper, we propose a generalized framework for the derivation of compact difference schemes for the approximation of an arbitrary order derivative. We formulate a methodology for calculating the approximation coefficients involving a large number of mesh points leading to very high-order formulas, e.g., 20th or 30th order. The main emphasis is on the first and second order derivative occurring in the Navier-Stokes equations governing fluid flow problems. Contrary to most studies on compact schemes in which the computational stencils employed in the discretization are narrow and consist of only a few mesh points K , in the presented paper we consider wide stencils that in practice are limited only by the overall number of the nodes in a computational grid. We analyze the features of such wide stencil based schemes in physical and spectral space. It is shown that the solutions obtained applying the schemes of the same order but involving different number of the mesh points characterize significantly different accuracy. This is presented by direct comparisons of the analytically derived formula for the leading terms of the truncation errors as well as in practical simulations. The test cases include a second order ordinary differential equation with variable coefficients, 1D Burgers equation and three typical benchmark problems used in Computational Fluid Dynamics (CFD), i.e., inviscid vortex advection, convected Taylor-Green vortices and a double jet flow. The obtained results are compared with available analytical solutions and the solutions obtained with the help of a pseudo-spectral method. We found that very high-order compact schemes reach the spectral accuracy characterized by the error decrease proportional to O (h K). In some cases, they show even higher accuracy, and moreover, reaching the limit of the machine precision they are substantially more resistant to the round-off errors. [ABSTRACT FROM AUTHOR]

In this paper, we prove that Crouzeix-Raviart finite elements of polynomial order p ≥ 5 , p odd, are inf-sup stable for the Stokes problem on triangulations. For p ≥ 4 , p even, the stability was proved by Á. Baran and G. Stoyan in 2007 by using the macroelement technique, a dimension formula , the concept of critical points in a triangulation and a representation of the corresponding critical functions. Baran and Stoyan proved that these critical functions belong to the range of the divergence operator applied to Crouzeix-Raviart velocity functions and the macroelement technique implies the inf-sup stability. The generalization of this theory to cover odd polynomial orders p ≥ 5 is involved; one reason is that the macroelement classes, which have been used for even p , are unsuitable for odd p. In this paper, we introduce a new and simple representation of non-conforming Crouzeix-Raviart basis functions of odd degree. We employ only one type of macroelement and derive representations of all possible critical functions. Finally, we show that they are in the range of the divergence operator applied to Crouzeix-Raviart velocities from which the stability of the discretization follows. [ABSTRACT FROM AUTHOR]

Anisotropic diffusion model has been established as one of the most significant image processing techniques, because it not only allows for space-variant filtering, but also realizes real direction-specific anisotropic smoothing of image structures. In this paper, an anisotropic nonlinear diffusion system is proposed for image enhancement and segmentation by taking into account the structure tensor constructed from the time-delay regularization and a nonlinear isotropic diffusion equation. Meanwhile, the diffusion system makes full use of the source term which increases the contrast and makes the image be less susceptible to noise, even in high level noise images. The source term also causes the restored image to tend to the binary image, which has proven to be an effective strategy for image segmentation. We analyze the proposed model in the view of the existence and uniqueness of the weak solution by using Galerkin's method. Some other theoretical analysis such as the maximum property is also discussed in the paper. With the appropriate and adaptive parameters, the anisotropic diffusion system achieves a better trade-off between computation time and image processing task. Furthermore, various image experiments illustrate that the proposed algorithm achieves much better efficiency and universality. [ABSTRACT FROM AUTHOR]

This paper is concerned with fast, parallel and numerically accurate pricing of two-asset American options under the Merton jump-diffusion model, which gives rise to a two-dimensional partial integro-differential complementarity problem (PIDCP) with a nonlocal two-dimensional integral term. Following method-of-lines approach, the solution to the PIDCP can be computed quite accurately by a robust numerical technique that combines Ikonen–Toivanen splitting with an alternating direction implicit scheme. However, we observed that computing the numerical solution with this technique becomes extremely time consuming, mainly due to the handling of the integral term. In this paper we parallelize this technique by applying a parallel fast Fourier transformation algorithm to all matrix-vector multiplications involving the huge and dense integral approximation matrix by exploiting its block Toeplitz with Toeplitz block structure. We also parallelize other computationally intensive steps of this technique by applying a recently developed parallel cyclic reduction algorithm for pentadiagonal systems. Our solutions computed on a graphics processing unit (GPU) using CUDA® platform are compared for accuracy with those available in the literature. It is observed that by solving the PIDCP parallelly we could bring down the computational times from several hours to a few seconds in certain cases in our experiments. [ABSTRACT FROM AUTHOR]

This paper considers the temporal discretization of an inverse problem subject to a time fractional diffusion equation. Firstly, the convergence of the L1 scheme is established with an arbitrary sectorial operator of spectral angle < π / 2 , that is the resolvent set of this operator contains { z ∈ C ∖ { 0 } : | Arg z | < θ } for some π / 2 < θ < π. The relationship between the time fractional order α ∈ (0 , 1) and the constants in the error estimates is precisely characterized, revealing that the L1 scheme is robust as α approaches 1. Then an inverse problem of a fractional diffusion equation is analyzed, and the convergence analysis of a temporal discretization of this inverse problem is given. Finally, numerical results are provided to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a lattice Boltzmann(LB) model based on the Cole-Hopf transformation is developed for the n -dimensional coupled Burgers' equations. Compared with the traditional LB model, the Cole-Hopf transformation is used to eliminate the nonlinear convection term in the Burgers' equations, such that simple diffusion equation can be obtained, then the LB model is proposed for the diffusion equation. Based on the Chapman-Enskog analysis, we can accurately recover the linear diffusion equation from this LB model, in which the variable θ in the linear diffusion equation and the gradient of the variable ∇ θ can be calculated conveniently by non-equilibrium distribution function without needing the finite-difference schemes. Then, the variables u in n -dimensional coupled Burgers' equations can be solved conveniently by θ and ∇ θ. Theoretically, this LB model can be used to solve any n -dimensional coupled Burgers' equations. Finally, some numerical experiments are conducted to test the present LB model, and the numerical results show that the model has the advantages in terms of computation, accuracy, stability. [ABSTRACT FROM AUTHOR]

In this paper we propose new Z-type nonlinear weights of the fifth-order weighted essentially non-oscillatory (WENO) finite difference scheme for hyperbolic conservation laws. Instead of employing the classical smoothness indicators for the nonlinear weights, we take the p th root of the smoothness indicators and follow the form of Z-type nonlinear weights, leading to fifth order accuracy in smooth regions, even at the critical points, and sharper approximations around the discontinuities. We also prove that the proposed nonlinear weights converge to the linear weights as p → ∞ , implying the convergence of the resulting WENO numerical flux to the finite difference numerical flux. Numerical examples are presented by comparing with other WENO schemes, such as WENO-JS, WENO-M and WENO-Z, to demonstrate that the proposed WENO scheme performs better in shock capturing. [ABSTRACT FROM AUTHOR]

In this contribution, we propose and analyze an efficient reduced-order finite element method for the parametrized Allen-Cahn equation. First, for a given parameter, the equation is discretized by a stabilized semi-implicit scheme in time and finite element method in space. Then, the reduced basis is constructed by using proper orthogonal decomposition (POD) to an ensemble of snapshots, which are numerical solutions of the full discrete problem at some time instances and a number of parameters over the parametric domain. The main novelty of this work is the error analysis of the reduced-order model. For the first time an error estimate is derived for the reduced-order solution of the parametrized Allen-Cahn equation. To the best of our knowledge, no published paper takes into account the impact of the diffusion parameter when proving bounds on the errors. Moreover, we propose a POD technique using time difference quotients as snapshots to improve the error estimates with respect to time step. Finally, we provide several numerical examples to verify the theoretical error estimates. • We propose a certified reduced order method for the parametrized Allen Cahn equation. • We present an error analysis including a specific use of time difference quotients. • We give numerical experiments to prove the efficiency of the strategy. [ABSTRACT FROM AUTHOR]

In this paper we provide guidelines for the construction of new high order conforming finite element exact sequences of subspaces in H 1 (Ω) , H (curl , Ω) , H (div , Ω) , and L 2 (Ω). They are meant for the design of stable and conservative mixed formulations of multiphysics systems combined with advanced numerical strategies. For instance, they support general polytopal meshes and/or allow trace interface constraints. These are appealing attributes for various challenging domains in computational practice, e.g. in hp -adaptivity, meshing of complex regions, multiscale simulations etc. Firstly, we consider standard finite element exact sequences of composite polynomial approximations inside the polytopal subdomains, based on conformal and regular sub-partitions of them. Over a partition of the mesh skeleton (faces and/or edges), we fix a sequence of trace spaces of piecewise polynomials. These trace spaces are supposed to form a hierarchy of lower dimensional exact sequences. Then, the trace-constrained subspaces only keep the non-vanishing trace components constrained by the given (coarser) trace spaces, but the internal components of the original local finite element spaces, having vanishing traces, are all kept. These internal (bubble) components may be richer to different extents when compared with the trace components with respect to internal mesh size, internal polynomial degree, or both. In principle, polynomial degree, mesh resolution and geometry are allowed to vary over different subdomains, but some interface compatibility properties are required. The operators over these trace-constrained exact finite element spaces commuting de Rham diagram are constructed in the context of projection-based interpolants. Some non-conventional examples of these sequences are described, illustrating the use of the specific finite element pairs in H (div , Ω) × L 2 (Ω). • Construction of FE spaces on polytopal meshes with trace interface constraints. • Divide-and-conquer approach combined with local bubble enrichment techniques. • Focus on advanced stable conservative mixed formulations of multiphysics systems. • Attributes for challenging tasks: hp-adaptivity, complex regions, multiscale. • Illustrating examples are used for porous media Darcy flows. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a highly scalable domain decomposition based solver for the fully implicit solution of subsurface flows in porous media with fractional derivative. In the proposed approach, a fully implicit scheme is proposed to relax the time step size by the stability condition, and the Newton–Krylov method is then applied to guarantee the nonlinear consistency of the resultant algebraic system. Moreover, to accelerate the convergence of the Newton iterations and be favorable for the scalability of the linear solver, the emphasis is placed on introducing the overlapping restricted additive Schwarz (RAS) preconditioners. Numerical experiments are presented to demonstrate the robustness and parallel scalability of the proposed fully implicit algorithm for some subsurface flow problems on a supercomputer. [ABSTRACT FROM AUTHOR]

In this paper, a new meshless symplectic and multi-symplectic scheme is developed for solving the coupled nonlinear Schrödinger system. The key feature of the method is the utilization of local radial basis function (RBF) approximation (LRBFA) method in spatial discretization. The LRBFA only requires solving a small linear system, thus it can circumvent the ill-conditioned problem and shape-parameter-sensitivity of global RBF method. The proposed method offers a high order of convergence and conserves the invariants well. Various numerical examples are presented to verify that the proposed method is valid for both uniform knots and random knots. [ABSTRACT FROM AUTHOR]

This paper proposes and analyzes a stabilized embedded discontinuous Galerkin (EDG) method for linear elasticity problems. A stabilized term is added in the scheme to ensure locking-free approximations. With a special designed interpolation, we prove the Stokes stable elements in space [ H 0 1 (Ω) ] d × L 0 2 (Ω) are also stable in [ H D 1 (Ω) ] d × L 2 (Ω) on general simplicial mesh, and therefore, we prove the underlying EDG method is robust in the sense that the derived a priori error estimates are optimal and uniform with respect to the Lamé constant λ. Numerical experiments are provided to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we present a calcium dynamics model that firmly orchestrates exchanges of calcium flux through intracellular/extracellular sources of calcium to investigate cellular activities and calcium homeostasis. The generalized two-dimensional space-time reaction-diffusion model is formulated and analyzed for a wide range of calcium flux. The analytical technique is applied to solve the model by use of the fractional integral transform and some special functions. The existence and uniqueness of the proposed model are discussed. Finally, the obtained results are presented graphically to show the significance of calcium flux by amalgamating the memory of cells and prove the efficiency of the method. • We developed a calcium dynamics model to investigate the interplay of calcium flux based on a reaction-diffusion equation. • We present the computational technique to derive the model's solution using integral transform and special functions. • The existence and uniqueness of the calcium dynamics model are discussed. • We estimated the concentration profile in the presence/absence of VGCC, ER leak, SERCA pump, NCX, and mitochondrial flux. • The modulation of calcium flux provides a noteworthy effect due to the intermediate memory of cells. [ABSTRACT FROM AUTHOR]

Droplet evaporation is a typical multiphase flow with various applications in many natural phenomena and industrial processes. Different numerical methods have been used to simulate this type of flow. Front-Tracking is one of the common methods in the study and simulation of multiphase flows. The Front-Tracking method requires very fine resolution when dealing with high density ratios with phase change. As a result, Front-Tracking method has rarely been used for solving problems at high-density ratios. In this paper, by using a two-step algorithm for moving the front, the Front-Tracking method is modified for problems with high-density ratios. Also, the fluid around the droplet is considered as a gas mixture, and the generated vapor is considered as a different component from the primary gas that surrounds the droplet. The results of the two-step algorithm are compared with the existing algorithm, and the available analytical results for a static droplet. The evaporation of a falling droplet is also studied, and the results of the two-step algorithm and common algorithm are compared. The effect of droplet deformation on the evaporation process for a falling is investigated. The effect of vapor properties on evaporation rate is also studied. The two-step algorithm is used for different three-dimensional cases, and static droplet evaporation is simulated. Results are compared with existing analytical results. Next, a falling droplet is modeled and the results are compared with experimental data. The effect of density ratio on falling droplet evaporation is also investigated. Finally, two and three-dimensional results are presented and compared for a specified case. For some cases, the conservation of mass and the dependence of results to grid resolution are presented which indicates the accuracy of the numerical method. [ABSTRACT FROM AUTHOR]

In this paper, based on a modified leap-frog scheme used in temporal direction and bilinear rectangular element applied in spatial direction, the superconvergence of Galerkin approximation are investigated for the cubic nonlinear Schrödinger equation with wave operator. The key issue to our analysis is to obtain the boundedness of the numerical solution in H 1 -norm, which is indeed different from the boundedness of the L ∞ -norm required in the previous literatures. The unconditionally superconvergence error estimates are obtained without any restrictions between time stepsize and spatial meshsize. Finally, some numerical results are provided to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper aims to address the material and spatial uncertainties in the stochastic electric field problem by proposing two efficient algorithms based on the stable nodal integration method (SNIM). The random variables/fields are used to describe the material and spatial uncertainties and a generalized stochastic perturbation-SNIM (GS-SNIM) method is developed to address the random-variable scenario while a random field-SNIM (RF-SNIM) method is developed to address the random-field scenario. Numerical examples are studied to fully verify the accuracy and efficiency of the proposed GS/RF-SNIM. The analysis results demonstrate that (1) the GS-SNIM is able to solve the random-variable based uncertainties through different orders expansion; (2) the RF-SNIM can calculate the expected values and variances of the stochastic electric field problem; and (3) the present methods produce accurate stochastic analysis results and significantly reduce the computational cost. • GS/RF-SNIM is proposed for stochastic electric field problems. • Both material uncertainty and spatial uncertainly are considered. • The GS/RF-SNIM can achieve high level of accuracy and efficiency. [ABSTRACT FROM AUTHOR]

This paper provides the global L 2 superconvergence of the tetrahedral quadratic finite element ‖ u h − u I ‖ 0 ≤ C h 4 ‖ u ‖ 4 , where u h and u I are the finite element approximation and the quadratic Lagrange interpolation of the exact solution u respectively. The standard analysis of the global L 2 superconvergence is combined with the weak estimate of the second type (WE2). However, the WE2 of the tetrahedral quadratic finite element has kept absent for a rather long time. To this end, we define the four-element-based uniform tetrahedral mesh, and then present the simplified weak estimate of the second type (SWE2), which simplifies the result of the WE2 and can also derive the global L 2 superconvergence. For the completeness, the proof of the WE2 can be found in the appendix. Finally, this supercloseness will be used to construct a post-processing that increases the order of approximation to the exact solution in L 2 norm. Numerical experiments are provided to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider a polynomial-augmented radial basis function collocation method (RBFCM) for solving fourth-order multi-dimensional elliptic partial differential equations (PDEs). Instead of choosing the RBF centres inside the problem domain, we use fictitious centres that can be located outside it to deal with the double boundary conditions of the fourth-order PDEs, and to also improve the computational accuracy. Furthermore, additional polynomial constraints applied to the approximation matrix of the RBFCM using fictitious centres, significantly alleviate the sensitivity of the method to the variation of the shape parameter, whereby it makes the RBFCM more practical for solving partial differential equations. We also apply the proposed method to a fourth-order elliptic PDE with random forcing terms (input data of the model). The input data are assumed to depend on a finite number of random variables at the collocation points. Numerical experiments are presented to demonstrate the effectiveness of the proposed method in terms of the selection of the shape parameter. In addition, the numerical results show that the proposed method achieves a higher accuracy than the conventional RBFCM. [ABSTRACT FROM AUTHOR]

In this paper, we utilise the physics-informed neural networks (PINN) combined with interpolation polynomials to solve nonlinear partial differential equations and for simplicity, the resulted neural network is termed as polynomial interpolation physics-informed neural networks (PI-PINN). Classically, the neural network is expressed as a power series by optimization of the coefficients to get an approximate solution of the partial differential equations (PDEs). Due to well-defined approximate properties of orthogonal polynomials, orthogonal polynomials are used to construct the neural network. Compared with PINN, PI-PINN clearly has a simple structure and is easy to be understood. We carry out some numerical experiments, including parabolic partial differential equations, hyperbolic partial differential equations and an application in fluid mechanics. By these investigations, it is further demonstrated that the PI-PINN structure is effective in solving nonlinear partial differential equations. Further, investigations on reverse problems are taken and accurate results are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we relax the equality constraints of the diffeomorphic image registration in Han and Wang (2020) [9] to equality and inequality constraints. Then, the mesh folding is eliminated. Moreover, a numerical algorithm for the proposed model is presented. Numerical tests are also performed to show that the proposed algorithm is competitive compared with some published image registration models. [ABSTRACT FROM AUTHOR]

A Rotated Lattice Boltzmann Flux Solver (RLBFS) with improved stability is proposed in this paper for the effective simulation of compressible flows with intense shock waves at high Mach numbers. Unlike the conventional LBFS or conventional Riemann solvers, which evaluate its mass, momentum and energy fluxes in the normal direction of each cell interface, the present RLBFS decomposes the outer normal direction of the cell interface into two perpendicular directions of the velocity difference and their counterparts. The fluxes in these two directions are respectively evaluated by the LBFS and the final fluxes at each interface are obtained by weighted summations of the two obtained fluxes. The matrix-based stability theory is applied to analyze the proposed solver by simulating the strong normal shock problem from Mach number 1.5 to 10. It reveals that the RLBFS has a smaller negative eigenvalue than the original LBFS, indicating that the RLBFS is more stable. The numerical accuracy of the RLBFS is evaluated by simulating the advection of density perturbation. The performance of the RLBFS is further demonstrated by simulating seven more compressible flows, including the transonic flow over RAE2822 airfoil, odd-even grid perturbation problem, a Mach 3 wind tunnel with a step, supersonic corner flow, double Mach reflection of a strong shock, shock-boundary layer interaction and hypersonic viscous flow around a blunt body. The obtained results agree well with the published data, showing that the RLBFS can effectively avoid the phenomenon of shock instability and thus has a potential for simulating practical compressible flows with intense shock waves at high Mach numbers. • A rotated lattice Boltzmann flux solver with improved stability is proposed. • A matrix-based stability theory is used to analyze the RLBFS and the LBFS. • Theoretical analysis show that the RLBFS is more stable than LBFS. • Numerical examples show the reliability and capability of the RLBFS. [ABSTRACT FROM AUTHOR]