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The stability, robustness, accuracy, and efficiency of space-time finite element methods crucially depend on the choice of approximation spaces for test and trial functions. This is especially true for high-order, mixed finite element methods which often must satisfy an inf-sup condition in order to ensure stability. With this in mind, the primary objective of this paper and a companion paper is to provide a wide range of explicitly stated, conforming, finite element spaces in four dimensions. In this paper, we construct explicit high-order conforming finite elements on 4-cubes (tesseracts); our construction uses tools from the recently developed 'Finite Element Exterior Calculus'. With a focus on practical implementation, we provide details including Piola-type transformations, and explicit expressions for the volumetric, facet, face, edge, and vertex degrees of freedom. In addition, we establish important theoretical properties, such as the exactness of the finite element sequences, and the unisolvence of the degrees of freedom. [ABSTRACT FROM AUTHOR]

In this paper the construction of C k basis functions on meshes with hanging nodes for hp -finite element methods is proposed. The construction is limited to axis-parallel, rectangular meshes, which can, however, be of any dimension. The basis functions are composed by using hierarchical tensor product shape functions which are defined on patches of so-called constraining nodes and combine univariate Hermite interpolation polynomials and Gegenbauer polynomials. The paper discusses how to integrate the constructed C k basis functions into the assembling process of stiffness matrices and load vectors by using connectivity matrices. Numerical examples illustrate the applicability of the proposed concept. [ABSTRACT FROM AUTHOR]

This paper employs a numerical approach to investigate the impact of liquidity shocks on European options in modeling markets. To accurately capture the behavior of European options under liquidity shocks, a coupled system of differential equations is employed, consisting of a degenerate parabolic equation and a diffusion-free equation. The primary focus is on developing and analyzing implicit-explicit methods for numerically simulating European option pricing, specifically considering the presence of liquidity shocks while ensuring the positivity of the solution. The paper also includes convergence analysis and establishes the discrete comparison principle for the developed methods. Numerical experiments are conducted using both uniform and nonuniform meshes to validate the theoretical findings, demonstrating the efficiency and accuracy of the proposed methods. [ABSTRACT FROM AUTHOR]

A novel higher-order multi-scale computational approach for efficient nonlocal gradient elasticity simulation of composite materials is proposed in this paper. Based on the use of decoupling technique, raw fourth-order nonlocal gradient equations with periodically oscillatory coefficients are decoupled as the new second-order multi-scale equations for reducing the continuity requirements when implementing multi-scale simulation. Next, a novel macro-micro coupled multi-scale computational model with higher-order corrected terms is rigorously devised for high-resolution multi-scale and nonlocal analysis of composite materials via multi-scale asymptotic analysis. Then, the local error analysis of higher-order multi-scale solutions in the point-wise sense illustrates that establishing higher-order asymptotic corrected terms plays a vital role in investigating the micromechanical mechanisms. Moreover, a rigorous global error estimation with an explicit rate of higher-order multi-scale solutions is first derived in the energy norm sense. In addition, an efficient multi-scale numerical algorithm is presented to effectively simulate nonlocal gradient elasticity problems of composite materials based on finite element method (FEM). And we also derive the convergence estimation of the proposed numerical algorithm. Finally, numerical experiments are conducted to gauge the efficiency and accuracy of the proposed higher-order multi-scale method. This paper offers a unified higher-order two-scale computational framework for enabling nonlocal gradient elasticity behavior simulation of composite materials. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose an Eulerian type of numerical finite difference approximation on structured grids for an advection dominated level set equation enhanced with a small curvature term. The proposed numerical scheme is based on upwind discretization in both the advection and the curvature term. This class of approximations is used in algebraic solvers, such as fast sweeping or fast marching methods, which respect the causality of solutions for which any information is propagating along characteristics, in order to solve the algebraic system in a finite number of computationally simple iterations or updates. An implicit numerical scheme for a time relaxed level set equation with non-zero right-hand side is used in this paper and solved by a nonlinear fast sweeping method. The results of the von Neumann stability analysis of the proposed scheme are presented, and numerical examples are used to verify the analysis. [ABSTRACT FROM AUTHOR]

Brain stroke is one of the most common causes of disability and death worldwide. In this paper, we present the numerical algorithm behind a new system for brain stroke detection: it relies on a novel imaging technique known as Electromagnetic Tomography (EMT). As compared to classical techniques, EMT has several advantages: it is noninvasive and, since it relies on nonionizing radiation, it is safe for the patient; it can be made compact and portable; it is easy to use and it has relatively low cost. The main drawback of EMT is its relatively poor spatial resolution due to restriction on the frequency spectrum. This paper describes the numerical challenges we had to overcome when designing our latest generation scanner. [ABSTRACT FROM AUTHOR]

We develop a monotone finite volume element scheme for the diffusion problem on arbitrary polygonal grids inspired by the scheme in the literature ([35]). The main contributions of this paper include three aspects. Firstly, some auxiliary lines or points are introduced for any polygonal element so that it can be partitioned into two or more triangles, and one novel type of its control volume is concomitantly designed. In this paper, we mainly discuss quadrilateral grids. Secondly, some piecewise linear finite element spaces are imported for the trial spaces. Fictitious elements, continuous extensions and the nonlinear two-point flux idea can all be inherited from the triangular case. Therefore, the monotonicity of the scheme is easily proved only by the assembling of some triangles' element stiff matrices. Thirdly, the agreeable adaptability for distorted grids is completely borned. The strict convexity restriction on the grids can also be removed, the concave or severely distorted polygonal (such as quadrilateral) grids are transformed as the convex and general regular triangles. Finally, numerical results confirm its monotonicity, accuracy and stability. [ABSTRACT FROM AUTHOR]

This paper deals with speeding up the convergence of a class of two-step iterative methods for solving linear systems of equations. To implement the acceleration technique, the residual norm associated with computed approximations for each sub-iterate is minimized over a certain two-dimensional subspace. Convergence properties of the proposed method are studied in detail. The approach is further developed to solve (regularized) normal equations arising from the discretization of ill-posed problems. The results of numerical experiments are reported to illustrate the performance of exact and inexact variants of the method on several test problems from different application areas. [ABSTRACT FROM AUTHOR]

This paper proposes a method to solve a 2D time-fractional parabolic equation on arbitrary domains using an RBF partition of unity procedure based on a non-uniform time-stepping framework. The proposed method involves partitioning the main domain into overlapping subdomains, computing local approximations on each subdomain, and joining them together by PU weights to obtain a global approximation. This approach enhances stability, reduces computational costs, and replaces an ill-conditioned linear system with several small well-conditioned linear systems. To handle the weak singularity near the initial time, a time-graded mesh is used to obtain a non-uniform time-stepping scheme, which increases the temporal convergence rate. The convergence analysis of the method is examined to obtain an error bound for the local approximation. Numerical tests are conducted to validate the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The Darcy's equation consists of the mass conservation equation and the Darcy's law that involves the hydraulic potential (or called pressure) and the fluid velocity, which governs the flow of an incompressible fluid through a porous medium. In this paper, we investigate the Legendre Galerkin spectral collocation least squares method for approximating the problem of Darcy flow in homogeneous medium and non-homogeneous medium, respectively. The proposed scheme can be solved the approximate solutions of the hydraulic potential and the average velocity of the fluid simultaneously. A symmetric positive definite coefficient matrix of the corresponding linear algebra equation is obtained by applying our scheme. Numerical examples are presented to validate the efficiency and accuracy of the proposed scheme. [ABSTRACT FROM AUTHOR]

This paper is devoted to a study of a history-dependent mixed hemivariational-variational inequality arising in contact mechanics. The contact problem concerns the deformation of a viscoelastic body with long memory, subject to a general friction law on one part of the boundary and a frictionless Signorini condition on another part of the boundary. The solution existence and uniqueness of the history-dependent mixed hemivariational-variational inequality based on a Lagrange multiplier approach are proved. Then, a fully discrete scheme is introduced and studied. The trapezoidal rule is used to approximate the integral in the history-dependent operator. For the spatial discretization, the linear finite elements are used to discretize the displacement field, and dual basis functions are used in the approximation of the Lagrange multiplier. Optimal order error estimates are derived for the displacement and the Lagrange multiplier under appropriate solution regularity assumptions. Numerical results are presented to illustrate the theoretical prediction of the convergence orders. [ABSTRACT FROM AUTHOR]

This paper proposes a low-cost, penalty parameter-free, and pressure-robust Stokes solver based on the enriched Galerkin (EG) method with a discontinuous velocity enrichment function. The EG method employs the interior penalty discontinuous Galerkin (IPDG) formulation to weakly impose the continuity of the velocity function. However, despite its advantage of symmetry, the symmetric IPDG formulation requires a lot of computational effort to choose an optimal penalty parameter and compute different trace terms. To reduce such effort, we replace the derivatives of the velocity function with its weak derivatives computed by the geometric data of elements. Therefore, our modified EG (mEG) method is a penalty parameter-free numerical scheme that has reduced computational complexity and conserves the optimal convergence orders. Moreover, we achieve pressure robustness for the mEG method by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. The theoretical results are confirmed through numerical experiments with two- and three-dimensional examples. [ABSTRACT FROM AUTHOR]

In this paper, a high order meshless numerical scheme based on the Generalized Finite Difference Method (GFDM) is proposed to analyze the anisotropic elliptic interface problem. The GFDM has an advantage in dealing with some anisotropic elliptic problems with complex interfaces, interface conditions with the jump of derivatives. Four problems with eight examples are provided to verify the existence of the good performance of the GFDM for anisotropic elliptic interface problems, including that the simplicity, accuracy, and stability in static and moving systems. Especially, GFDM has the tolerance of the complexity of interface shape, the interface movement and the large jump. [ABSTRACT FROM AUTHOR]

In this paper, we present the coupled physics-informed neural networks (CPINNs) for solving the two phase magnetohydrodynamic flows which is governed by Cahn–Hilliard equation and MHD equations. Central to this method is coupling multi-feedforward neural networks to solve variables in parallel. Specifically, CPINNs embed the model into the loss function of the neural networks and train neural networks to obtain the optimal parameters, among which the resulting networks satisfy differential operators, initial and boundary conditions. Based on the idea of small sample learning (SSL), we utilize few sample data during the training process, which can alleviate the calculated costs and avoid grid generation. Additionally, we rigorously demonstrate the existence and convergence of the resulting neural networks. Finally, the theoretical results are verified via several numerical experiments. [ABSTRACT FROM AUTHOR]

This paper aims to reconstruct the initial condition of a hyperbolic equation with an unknown damping coefficient. Our approach involves approximating the hyperbolic equation's solution by its truncated Fourier expansion in the time domain and using the recently developed polynomial-exponential basis. This truncation process facilitates the elimination of the time variable, consequently, yielding a system of quasi-linear elliptic equations. To globally solve the system without needing an accurate initial guess, we employ the Carleman contraction principle. We provide several numerical examples to illustrate the efficacy of our method. The method not only delivers precise solutions but also showcases remarkable computational efficiency. [ABSTRACT FROM AUTHOR]

In this paper, we generalize a fast Fourier transforms (FFTs) based preconditioner and propose a novel α -absolute value preconditioner for all-at-once systems from heat equations. Our preconditioner is symmetric positive definite and can also be computed efficiently using fast Fourier transforms or discrete sine transforms (DSTs). Furthermore, we prove that the corresponding preconditioned matrix has a unitary-plus-low-rank decomposition, and the eigenvalues of the preconditioned matrix are clustered at ±1, which leads to fast convergence rate of the preconditioned MINRES method. Finally, numerical examples on all-at-once systems from heat equations with different discretization scheme demonstrate the effectiveness of our new preconditioner. [ABSTRACT FROM AUTHOR]

The paper is concerned with the mathematical theory and numerical approximation of systems of partial differential equations (pde) of hyperbolic, pseudo-parabolic type. Some mathematical properties of the initial-boundary-value problem (ibvp) with Dirichlet boundary conditions are first studied. They include the weak formulation, well-posedness and existence of traveling wave solutions connecting two states, when the equations are considered as a variant of a conservation law. Then, the numerical approximation consists of a spectral approximation in space based on Legendre polynomials along with a temporal discretization with strong stability preserving (SSP) property. The convergence of the semidiscrete approximation is proved under suitable regularity conditions on the data. The choice of the temporal discretization is justified in order to guarantee the stability of the full discretization when dealing with nonsmooth initial conditions. A computational study explores the performance of the fully discrete scheme with regular and nonregular data. [ABSTRACT FROM AUTHOR]

This article introduces a unified mixed finite element framework based on a saddle-point formulation that applies to time-dependent fourth order linear and nonlinear problems with clamped, simply supported, and Cahn-Hilliard type boundary conditions. The classical mixed formulations lead to large matrix systems that demand huge storage and computational time making the schemes expensive, especially for the time-dependent problems. The proposed scheme circumvents this by employing biorthogonal basis functions that lead to sparse and positive-definite systems. The article discusses a mixed finite element method for the biharmonic problem and the time-dependent linear and nonlinear versions of the extended Fisher-Kolmogorov equations equipped with the aforementioned boundary conditions. The wellposedness of the scheme is discussed and a priori error estimates are presented for the semi-discrete and fully discrete finite element schemes. The numerical experiments validate the theoretical estimates derived in the paper. [ABSTRACT FROM AUTHOR]

This paper presents and analyzes a finite volume scheme for a fully degenerate parabolic system modeling the compressible two-phase flow, such as water-gas, problem in porous media considering the effects of temperature. The problem is written in terms of the phase formulation, i.e. the saturation of one phase, the pressure of the second phase and the temperature are primary unknowns. We use a first-order time implicit Euler scheme. The spatial discretization uses a fully coupled fully implicit cell-centered two-point flux approximation scheme where upwinding of the flux across an interface between two control volumes is performed for each fluid phase. Under physically relevant assumptions on the data, we prove the maximum principle for both saturation and temperature. We obtain a priori estimates and we show that the discrete problem is well posed. Also, we exploit various a priori estimates as well as compactness arguments to establish the convergence of the numerical scheme toward the weak solution. The open-source platform DuMu X was employed for the implementation of the developed algorithm based on the above scheme. Numerical result are presented to demonstrate the efficiency of this scheme. The test addresses the evolution in 2D of CO 2 injection into a layered aquifer. The algorithm's convergence with first order is validated by our numerical results, which support the theoretical results. To our best knowledge, this is the first convergence result of a finite volume scheme in the case of nonisothermal immiscible compressible two-phase flow in porous media. [ABSTRACT FROM AUTHOR]

In this paper, we propose a decomposition formula for the fluid-solid interaction scattering problem. By the decomposition of the displacement, the corresponding problem is transformed into a coupled boundary value problem. For the two dimensional case, the coupled boundary value problem is about two scalar Helmholtz equations, and the other one is a scalar Helmholtz equation and a vector Helmholtz equation for the three dimensional. Then the coupled boundary value problem is solved by the method of fundamental solution. At last we give some examples to show the effectiveness of the present method in two and three dimensional case. A benchmark example is available for checking the feasibility and accuracy of the presented method. In the numerical examples, a part of exact solution for the boundary value problem is including for checking the accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we propose a second-order adaptive DLN algorithm with different subdomain variable time steps for the 3D closed-loop geothermal system. Theoretically, the stability and convergence of the algorithm in the case of variable time steps are analyzed. In numerical experiments, the DLN algorithm is combined with an adaptive algorithm to verify not only its effectiveness and second-order convergence under any arbitrary sequence of time steps but also to improve computational efficiency. We overcome the difficulty of 3D programming, and verify the applicability and accuracy of the proposed algorithm by simulating the flow characteristics and heat transfer of the closed-loop geothermal system with 3D convection. [ABSTRACT FROM AUTHOR]

The advection-diffusion equation is fundamental to modeling various transport phenomena, including the distribution of chemical species in surface or groundwater flow. In cases where the concentration at the source is unknown, inverse problem formulations are required to estimate the desired states by assimilating concentration monitoring data from specific points along the watercourse using a Bayesian approach. This paper proposes a combination of the Eulerian-Lagrangian method and the meshless method of fundamental solutions to solve the advection-diffusion equation. Moreover, the method was used as an evolution model for the sequential importance resampling particle filter algorithm to reconstruct the time-dependent inlet pollutant concentration in inverse problems. Numerical smooth and discontinuous inlet function results show that the particle filter - Eulerian-Lagrangian method of fundamental solutions combination can reconstruct inlet concentration time series. • Verification and validation of ELMFS solution with analytical, FDM and experimental data. • The ELMFS was coupled with sampling importance resampling particle filter for the solution of inverse problem. • The proposed framework is reliable and robust in reconstructing the inlet boundary conditions. [ABSTRACT FROM AUTHOR]

The efficient approximation of parametric PDEs is of tremendous importance in science and engineering. In this paper, we show how one can train Galerkin discretizations to efficiently learn quantities of interest of solutions to a parametric PDE. The central component in our approach is an efficient neural-network-weighted Minimal-Residual formulation, which, after training, provides Galerkin-based approximations in standard discrete spaces that have accurate quantities of interest, regardless of the coarseness of the discrete space. [ABSTRACT FROM AUTHOR]

The classic Landweber iteration method is widely used to solve ill-posed problems, and is proven to be very efficient. However the classic Landweber iteration method has the drawback of over smoothness near the cusps or jumps. Nowadays fractional regularization method becomes popular to overcome over-smoothness. In this paper we propose a fractional Landweber iteration method to solve a two-dimensional time-fractional inverse heat conduction problem. We give the selection of iterations under both a priori and a posteriori rules, and give the proof of error estimates between the exact solution and its regularization approximation. Further more, numerical results are included to verify the effectiveness of the proposed method. In experiment, we first calculate the direct problem of the heat conduction problem using the finite difference method; then we add random noise to the data and get the disturbed data; at last we compute the numerical approximate solution of the inverse problem with disturbed data by the proposed method. [ABSTRACT FROM AUTHOR]

As we know that, the parameterized Uzawa (PU) method can be very efficient when used to solve the standard saddle point problem, especially, when we have good and accurate estimation of preconditioned Schur complement matrix. In this paper, by taking full use of the special structure of coefficient matrix arising from complex Helmholtz equations, two types of optimized PU (OPU) methods are discussed theoretically and experimentally. Specifically, the convergence factors of these two OPU methods are less than 0.172, and the optimal result of first OPU method can reach 0.0396 with σ 1 ≥ σ 2 > 0 , which is currently the best theoretical result in the literatures. Moreover, the second OPU method has better computational advantages compared with the first OPU method as it avoids the inverse computation of W at each step, indicating the less CPU time will be costed by the second OPU method. In addition, the optimal parameters involved in our algorithms consist of constants, reducing the computational complexity associated with parameter selection. Finally, numerical results are given, not only show the effectiveness of OPU methods, but also confirm the rationality of theoretical analysis. [ABSTRACT FROM AUTHOR]

Multiplicative noise removal is a challenging image processing problem. In this paper, we study a fractional-order nonlinear cross-diffusion system to denoise images corrupted by multiplicative Gamma noise. The system is coupled by two nonlinear, partial differential equations and gray level information, fractional derivative operator is considered in diffusion process to remove high level noise effectively while preserve structure details for texture images and SAR images. We prove the existence of weak solutions and other theoretical properties of the fractional-order cross-diffusion system under some hypotheses. In the numerical aspect, the model is implemented by virtue of the discrete fractional Fourier transform in the frequency domain. Experimental results show that the new system has a good compromise between multiplicative noise removal and structure preservation when compared to other well-known denoising methods. [ABSTRACT FROM AUTHOR]

In this paper, based on the Lagrange multiplier method, we construct a maximum principle preserving scheme for the time-fractional Allen-Cahn equation of 2- α (0 < α < 1) order. The correction energy of this scheme is increased by a term compared to the original energy, which is O (τ α). We prove that our scheme is unconditionally stable related to the corrected energy and verify the convergence, maximum principle, and energy conservation properties of the algorithm through numerical examples. We also find that the larger the α , the faster the evolution of the time-fractional Allen-Cahn equation. [ABSTRACT FROM AUTHOR]

In this paper, we consider a finite element approximation of an optimal control problem governed by a time fractional wave equation. We first prove the well-posedness and regularity of the optimal control problem. Furthermore, we discuss the fully discrete scheme and analyze its stability and a priori error estimate. Numerical experiments are given to illustrate the theoretical findings. • The weak solution to the time fractional wave equation is studied under low regular data. • The well-posedness and regularity of the optimal control problem are proven. • Numerical schemes exist the first-order and second-order accuracy in time and space respectively. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the modified Navier-Stokes-Cahn-Hilliard equations with a data assimilation term to utilize the information of the observed data. Based on the idea of feedback control, this term nudges the solutions to the observed data sampled from the reference process. By utilizing the Crank-Nicolson formula and the scalar auxiliary variable approach, we introduce an efficient numerical scheme for the modified Navier-Stokes-Cahn-Hilliard equations. Properties of the mass conservation and unconditional energy stability are proved. We explore the robustness and efficiency of the proposed scheme with various experiments. [ABSTRACT FROM AUTHOR]

This paper explores the utilization of randomized SVD (rSVD) in the context of kernel matrices arising from radial basis functions (RBFs) for the purpose of solving interpolation and Poisson problems. We propose a truncated version of rSVD, called trSVD, which yields a stable solution with a reduced condition number in comparison to the non-truncated variant, particularly when manipulating the scale or shape parameter of RBFs. Notably, trSVD exhibits exceptional proficiency in capturing the most significant singular values, enabling the extraction of critical information from the data. When compared to the conventional truncated SVD (tSVD), trSVD achieves comparable accuracy while demonstrating improved efficiency. Furthermore, we explore the potential of trSVD by employing scale parameter strategies, such as leave-one-out cross-validation and effective condition number. Then, we apply trSVD to solve a 2D Poisson equation, thereby showcasing its efficacy in handling partial differential equations. In summary, this study offers an efficient and accurate solver for RBF problems, demonstrating its practical applicability. The code implementation is provided to the scientific community for their access and reference. [ABSTRACT FROM AUTHOR]

In this paper, we present a new model as an alternative to the classical 3D Richards model for the description of water flow in shallow aquifers. The new model is designed to achieve three goals. First, it provides a good approximation to the Richards model over a wide range of time scales. More specifically we show that both models characterize a flow with the same dominant components when the ratio of the horizontal length to the depth of the aquifer is small. Second, the new model accurately describes the velocity field. In particular, it is not based on the Dupuit hypothesis, which is often used in the context of shallow aquifers. This allows the new model to be well accurate even in the presence of wells and in aquifers with variable bedrock. Third, the new model can be viewed as a coupling of numerous 1D vertical Richards problems with a 2D elliptic one. In practice, this coupling is treated numerically using a Picard fixed point scheme, avoiding the need to solve any three-dimensional problem. This results in a significant reduction in computational cost compared to solving the 3D Richards problem directly. [ABSTRACT FROM AUTHOR]

In this paper, we devise a stochastic extragradient-like algorithm incorporated with inertial terms, which requires a single projection onto our feasible set and employs a stochastic approximation version of an Armijo-type line search scheme along a feasible direction, for solving pseudomonotone stochastic variational inequalities. In the algorithm, two different stepsize strategies are employed to update steplength sequences without using the prior knowledge of the Lipschitz constant of involved operators. In the process of the stochastic approximation, we iteratively reduce the variance of stochastic errors. The almost sure convergence, the complexity analysis, and rates are provided in a dimensional space under reasonable conditions. Finally, some numerical experiments with graphical illustrations are reported to demonstrate the applicability and the efficiency of our algorithm in comparison with some projection type methods in the literature. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose an efficient numerical scheme to deal with heat transfer problems in pipe flow with a large length/diameter ratio. The generalized finite difference method (GFDM) is combined with a space stepping algorithm (SSA) for the solution. The SSA divides the solution domain into a number of overlapping sections in the length direction of the pipe. Due to the large Peclet number, the heat transport process is dominated by advection, which allows an approximate boundary condition to be applied at the downstream cross section. The problem is then solved section by section. Using the uniform distributed points, the solution matrix for each section does not change, and only the right hand side needs to be refreshed for the changing boundary conditions. This leads to a highly efficient scheme for problems with a large pipe length. To show the accuracy of the numerical scheme, a problem with available analytical solution is studied. Then the method is applied to problems of single pipe with different pipe wall temperature and flow Reynolds number, and concentric annular pipe. Numerical results confirm the stability and efficiency of the GFDM-SSA. The method can be applied to many real world transient heat transfer problems in which the pipe is heated or cooled along its length with arbitrary wall temperature for heat exchange and other purposes. [ABSTRACT FROM AUTHOR]

In this paper, we present an improved dynamic analysis method to address plane elasticity problems. Distributed sources are employed in a boundary-free mesh-based approach known as Boundary Distributed Source (BDS). This method, which builds upon a modified Method of Fundamental Solutions (MFS), aims to eliminate singularities without the reliance on fictitious boundaries. Distributed sources are strategically positioned directly on the real boundary, centered over disks. The method's efficiency and simplicity are highlighted by its ability to operate without the need for precise data concerning adjacent points. The dual reciprocity method (DRM) is utilized to approximate the non-homogeneous inertial term. Furthermore, the present formulation investigates several factors that influence the convergence of the proposed method, including the number of collocation points, internal points, the radius of the circular disk, and the time step size. Based on the numerical results, it is evident that the method exhibits remarkable accuracy and efficiency when applied to plane elasticity problems including forced, harmonic, and free analyses. As compared to conventional numerical methods for dynamic analysis, the method outperforms in both accuracy and convergence rate. [ABSTRACT FROM AUTHOR]

In this paper we use the weighted discontinuous Galerkin finite element method to solve the dual-porosity-Brinkman coupling model. The dual-porosity model describes fluid flow in a dual-porosity medium composed of matrix and microfractures, while the Brinkman equation describes fluid flow in hydraulic fractures or conduits containing "debris filling" or "proppant filling". The two models are coupled by the four physically effective interface conditions, which are constructed based on the fundamental properties of the dual-porosity model and the Brinkman-Darcy model. We provide a weighted discontinuous discrete scheme for the model and derive the optimal error estimates. Through numerical experiments, we verify the properties of the model and the advantages of the numerical method, such as the optimal convergence rate of the numerical solution and the simulation of the flow characteristics around hydraulic fractures and conduits. [ABSTRACT FROM AUTHOR]

This paper investigates the stabilization of Euler-Bernoulli beam systems modeled by a partial differential equation (PDE) with unknown time-varying disturbance. An iterative learning controller is designed using only boundary state feedback to realize the vibration control subject to unknown boundary disturbance. The well-posedness for the closed-loop system is given by the operator semigroup theory. Furthermore, the exponentially stable for the closed-loop system is proved by the Lyapunov method. The comparisons with existing results are made to demonstrate the effectiveness and advantages of the proposed boundary iterative learning control method. [ABSTRACT FROM AUTHOR]

In this paper, staggered discontinuous Galerkin (SDG) approximation for the Stokes problem is developed in a 2D square domain. The square domain is discretized by using rectangular meshes, which allow us easy identification of elements and hence a simple implementation of the SDG methods. By introducing proper function spaces and interpolations for the velocity vector field, its gradient, and the pressure, some required inf-sup conditions of our SDG method are shown. Then, thanks to the inf-sup conditions, the optimal convergence of our SDG approximation to the Stokes solution is obtained with respect to the polynomial order. Numerical results are included as well to validate the performance of our proposed SDG method for the Stokes problem. [ABSTRACT FROM AUTHOR]

Numerical studies of flow in square enclosures attain great interest due to their wide applications in engineering, such as in the meteorology, navigation, mechanology and mining, etc. In this paper, the solutions are obtained to the square lid driven cavity describing classical mixed convection in the presence of an inclined magnetic field, chemical radiation and heat absorption/generation. The problem is mathematically reduced into Navier-Stokes equations. The Coiflet wavelet homotopy approach, a newly suggested computational technique, is then used to solve these nonlinear partial differential equations with their non-homogeneous boundary conditions. The numerical solutions for streamlines, isotherms, iso-concentrations contours, Nusselt number are carefully determined. The solutions so obtained are analyzed through various plots to demonstrate the effects of several physical parameters such as Reynolds numbers (Re), Richardson number (Ri), Grashof number (Gr), Hartman number (Ha), thermal radiation parameter (N R), chemical reaction coefficient (K), heat generation/absorption coefficient (Q h), the angle of inclination of magnetic field (γ), slipping parameter (λ) and Schmidt number (Sc) and their physical insights are also reported. Rigid comparison of with previous studies shows excellent computational effectiveness of newly proposed method. Additionally, according to the results, our recently proposed approach has superior characteristics and strong nonlinear processing capabilities, which render it more appropriate than the Coiflet wavelet analysis method and the traditional homotopy analysis method for tackling complex nonlinear problems. [ABSTRACT FROM AUTHOR]

The Maxwell polynomial chaos Debye model uses the parameter distribution in the Debye model to represent the Cole-Cole dispersive mechanism. In this paper, a leap-frog nodal discontinuous Galerkin (DG) method is studied for solving this model. Under the Courant-Friedrichs-Lewy (CFL) condition, the stability of the fully discrete DG scheme is demonstrated and its error estimate is derived. Numerical examples are provided for supporting the correctness of the theoretical analysis and the effectiveness of our proposed method. [ABSTRACT FROM AUTHOR]

This paper introduces general methodologies for constructing closed-form solutions to linear constant-coefficient partial differential equations (PDEs) with polynomial right-hand sides in two and three spatial dimensions. Polynomial solutions have recently regained significance in the development of numerical techniques for evaluating volume integral operators and also have potential applications in certain kinds of Trefftz finite element methods. The equations covered in this work include the isotropic and anisotropic Poisson, Helmholtz, Stokes, linearized Navier-Stokes, stationary advection-diffusion, elastostatic equations, as well as the time-harmonic elastodynamic and Maxwell equations. Several solutions to complex PDE systems are obtained by a potential representation and rely on the Helmholtz or Poisson solvers. Some of the cases addressed, namely Stokes flow, Maxwell's equations and linearized Navier-Stokes equations, naturally incorporate divergence constraints on the solution. This article provides a generic pattern whereby solutions are constructed by leveraging solutions of the lowest-order part of the partial differential operator (PDO). With the exception of anisotropic material tensors, no matrix inversion or linear system solution is required to compute the solutions. This work is accompanied by a freely-available Julia library, ElementaryPDESolutions.jl, which implements the proposed methodology in an efficient and user-friendly format. [ABSTRACT FROM AUTHOR]

This paper is devoted to develop a new energy dissipation scheme by finite element method (FEM) for a nonlinear Schrödinger equation with wave operator (NLSW) and investigate its superconvergence behavior through introducing a new second order discrete approximation with respect to time. The new scheme not only guarantees the mass and energy stable but also is efficient in achieving the numerical solution at the last point. Also, an application to other general nonlinear equations with wave operator is discussed. On the one hand, the boundedness of the solution about the numerical equation is deduced and as a result, the existence and uniqueness of the solution are obtained. On the other hand, the superconvergence error estimate is derived and some numerical results are given to confirm the theoretical predictions. [ABSTRACT FROM AUTHOR]

In this paper, elliptic control problems with the integral constraint on the gradient of the state and the box constraint on the control are considered. The optimality conditions for the problem are proved. To numerically solve the problem, a finite element duality-based inexact majorized accelerated block coordinate descent (FE-dABCD) algorithm is proposed. Specifically, both the state and the control are discretized by piecewise linear functions. An inexact majorized ABCD algorithm is employed to solve the discretized problem via its dual, which is a multi-block unconstrained convex optimization problem, but the primal variables are also generated in each iteration. Thanks to the inexactness of the FE-dABCD algorithm, the subproblems at each iteration are allowed to be solved inexactly. For the smooth subproblem, we use the preconditioned generalized minimal residual (GMRES) method to solve it. For the two nonsmooth subproblems, one of them has a closed form solution through introducing an appropriate proximal term, and another one is solved by the line search Newton's method. Based on these efficient strategies, we prove that our proposed FE-dABCD algorithm enjoys O (1 k 2 ) iteration complexity. Moreover, to make the algorithm more efficient and further reduce its computation cost, based on the mesh-independence of ABCD method, we propose an FE-dABCD algorithm with a warm-start strategy (wFE-dABCD). Some numerical experiments are done and the numerical results show the efficiency of the FE-dABCD algorithm and wFE-dABCD algorithm. [ABSTRACT FROM AUTHOR]

Two key challenges in optimal control include efficiently solving high-dimensional problems and handling optimal control problems with state-dependent running costs. In this paper, we consider a class of optimal control problems whose running costs consist of a quadratic on the control variable and a convex, non-negative, piecewise affine function on the state variable. We provide the analytical solution for this class of optimal control problems as well as a Hopf-type representation formula for the corresponding Hamilton-Jacobi partial differential equations. Finally, we propose efficient numerical algorithms based on our Hopf-type representation formula, convex optimization algorithms, and min-plus techniques. We present several high-dimensional numerical examples, which demonstrate that our algorithms overcome the curse of dimensionality. We also describe a field-programmable gate array (FPGA) implementation of our numerical solver whose latency scales linearly in the spatial dimension and that achieves approximately a 40 times speedup compared to a parallelized central processing unit (CPU) implementation. Thus, our numerical results demonstrate the promising performance boosts that FPGAs are able to achieve over CPUs. As such, our proposed methods have the potential to serve as a building block for solving more complicated high-dimensional optimal control problems in real-time. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the convergence rates of an implicit-explicit (IMEX) difference scheme for solving a two-dimensional partial integro-differential equation (PIDE) with an inexact boundary which arises in option pricing with stochastic intensity jumps. First, the IMEX scheme is proposed to solve the two-dimensional PIDE and its inexact boundary governed by a one-dimensional PIDE. Then the second-order convergence rates of the IMEX scheme for the main PIDE are proved for both time and space based on the second-order convergence analysis in the discrete H 1 -norm of the IMEX scheme for the boundary PIDE. Numerical examples are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In Beirão da Veiga et al. 2019, the authors proposed a virtual element method for domains with fixed curved boundary that tends to be locally straight when the discretisation gets finer and finer. In this paper, we properly modify this method so that it can also deal with curved mesh edges which are not necessarily located along boundaries and do not tend to be straight when refining the mesh. To achieve this goal, we assume that curved edges are described by polynomials and we increase the dimension of the virtual element space used in Beirão da Veiga et al. 2019. In the numerical experiments, we compare these two methods. Furthermore, we apply the proposed approach to the benchmark "TEAM 25" problem, an optimal shape design problem in magnetostatics characterised by curved edges. [ABSTRACT FROM AUTHOR]

This paper analyzes a parareal approach based on fractional-step methods for the nonstationary Navier-Stokes equations. As an efficient parallel computing framework, the coarse propagator often determines the performance of the parareal algorithm. We present a parareal algorithm using the fractional-step method, a very efficient time discrete scheme for the Naiver-Stokes equations, as the coarse propagator for the Navier-Stokes equations. Then we give the specific stability and convergence analysis of this specific parareal algorithm. Finally, numerical experiments are done to show efficiency and illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper focuses on investigating a second-order time parallel decoupled algorithm for the mixed Stokes/Darcy model. The main objective of this algorithm is to deal with the issue of low computational efficiency in solving the coupling problem directly. Our scheme not only reduces the computation scale through the decoupling method, but also realizes the parallelism in the time direction through parareal method. Furthermore, we implement a second-order spectral deferred correction method to improve accuracy. For some small time step size, we prove the stability and convergence results of the time parallel decoupled algorithm. Finally, we provide three numerical tests to demonstrate the reliability and computational efficiency of our algorithm. [ABSTRACT FROM AUTHOR]

In this paper, according to the element discretization characteristic of the geometric model, a complex function theory based transformation boundary element method is proposed to perform the steady-state thermal analysis of plate structures. The numerical implementation of the proposed method can be attributed to a numerical integration problem in element analysis, which is mainly divided into the following two steps: firstly, a bidirectional transformation method based on distance function and complex function is proposed to remove the weak singularities of the fundamental solution in integral equation, which is suitable for the weakly singular integrals of narrow strip discrete elements; secondly, a dual sinh method based on complex poles is developed to remove the influence of the nearly singular integrals in the analysis of the plate structures. With this method, the numerical integration in the integral equation can be accurately evaluated, which ensures the accurate calculation of the thermal physical variables of the plate structures. Several numerical examples are presented and discussed to prove the accuracy and versatility of the presented method. The results show that the presented method has the superiority of dealing with plate structure problems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and analyze a lowest-order locking-free weak Galerkin (WG) finite element scheme for the grad-div formulation of linear elasticity problems. The scheme uses linear functions in the interior of mesh elements and constants on edges (2D) or faces (3D), respectively, to approximate the displacement. An H (d i v) -conforming displacement reconstruction operator is employed to modify test functions in the right-hand side of the discrete form, in order to eliminate the dependence of the L a m e ´ parameter λ in error estimates, i.e., making the scheme locking-free. The method works without requiring λ ‖ ∇ ⋅ u ‖ 1 to be bounded. We prove optimal error estimates, independent of λ , in both the H 1 -norm and the L 2 -norm. Numerical experiments validate that the method is effective and locking-free. [ABSTRACT FROM AUTHOR]

In this paper, a simple preconditioner with two parameters (SPTP) is introduced for solving a class of saddle-point problems. The SPTP preconditioner is block upper triangular and does not contain the approximate matrix of the Schur complement matrix. Theoretical analyses on the sufficient conditions under which the SPTP iterative sequence converges to the unique solution, are given in detail. Convergence conditions in some special cases are also analyzed. Moreover, detailed analyses on the theoretical optimal parameters and corresponding optimal convergence factor of the SPTP method are given. Some numerical examples show that the SPTP iterative method outperforms several SOR-type methods by using theoretical optimal parameters simultaneously. Some newly proposed methods are also compared with the SPTP iterative method by using the experimental optimal parameters simultaneously, and the SPTP iterative method is still advantageous, from the point of view of iteration step and CPU time. Moreover, we also investigate relatively practical values of the parameters since searching for the optimal parameters will waste a lot of time in practice. [ABSTRACT FROM AUTHOR]