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In this paper, we consider efficient and energy-stable numerical schemes for solving a modified phase-field crystal model with a strong nonlinear vacancy potential. By combining the modified exponential SAV approach and the relaxed SAV approach, we introduce a new auxiliary variable to reformulate the model. We adopt the backward Euler formula and the second-order backward difference formula (BDF2) to develop the first- and second-order time-accurate schemes, respectively. The energy dissipation law can be proved for all proposed schemes. In each time step, the computation is totally decoupled. Non-local variables can be explicitly updated and the local variables can be computed by solving an elliptic type equation with constant coefficients. Numerous numerical experiments in 2D and 3D are carried out to show the accuracy and energy stability of the proposed schemes. • Energy stable schemes are proposed for the modified phase-field crystal model with a strong nonlinear vacancy potential. • The developed method is very easy to implement because all variables are decoupled. • The mass conservation and energy stability are proved analytically. • Our proposed scheme works well for numerical simulations in two-dimensional and three-dimensional. [ABSTRACT FROM AUTHOR]

Superconvergence error analysis is investigated of a mixed finite element method (MFEM) for the time-dependent natural convection problem. The backward Euler scheme is adopted for the discrete in time, while the Bernadi-Rangel element is used to approximate the velocity u and the pressure p , and the bilinear element to the temperature T in space, respectively. Combined with the characters of the chosen elements and skillful techniques, the superclose results of order O (h 2 + τ) for u , T in the H 1 -norm and p in L 2 -norm are deduced rigorously, where h is the subdivision scale and τ , the time step. Further, the global superconvergence results are obtained through the interpolation postprocessing technique. We mention that in the high accuracy results obtained in this paper, the restriction between the h and τ is removed by the space-time splitting skill. Finally, numerical tests are also established to verify the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, an efficient modified method of characteristics with adjust advection (MMOCAA) domain decomposition method (DDM) is analyzed for solving advection diffusion equations. On each non-overlapping block divided sub-domains, we take two half time steps along x -direction and y -direction, to solve the numerical solutions on each sub-domain at every time interval, respectively. The intermediate numerical solutions in the interiors of sub-domains are first computed by redefining three mass fluxes. Then we compute the other intermediate interior numerical solutions by the weighted scheme from the previous modified solutions. Finally, the interior solutions are computed by the splitting implicit solute-flux coupled scheme while the interface fluxes are computed explicitly by the local multi-point weighted scheme. The conservation, stability and convergence of our schemes are proved strictly. Numerical experiments are presented to illustrate the theoretical results and efficiency. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with Trefftz discretizations of the time-dependent linear wave equation in anisotropic media in arbitrary space dimensional domains Ω ⊂ R d (d ∈ N). We propose two variants of the Trefftz DG method, define novel plane wave basis functions based on rigorous choices of scaling transformations and coordinate transformations, and prove that the corresponding approximate solutions possess optimal-order error estimates with respect to the meshwidth h and the condition number of the coefficient matrices, respectively. Besides, we propose the global Trefftz DG method combined with local DG methods to solve the time-dependent linear nonhomogeneous wave equation in anisotropic media. In particular, the error analysis holds for the (nonhomogeneous) Dirichlet, Neumann, and mixed boundary conditions from the original PDEs. Furthermore, a strategy to discretize the model in heterogeneous media is proposed. The numerical results verify the validity of the theoretical results, and show that the resulting approximate solutions possess high accuracy. [ABSTRACT FROM AUTHOR]

In the seminal paper of Bank and Weiser (1985) [17] a new a posteriori estimator was introduced. This estimator requires the solution of a local Neumann problem on every cell of the finite element mesh. Despite the promise of Bank–Weiser type estimators, namely locality, computational efficiency, and asymptotic sharpness, they have seen little use in practical computational problems. The focus of this contribution is to describe a novel algorithmic approach to constructing hierarchical estimators of the Bank–Weiser type that is designed for implementation in a modern high-level finite element software with automatic code generation capabilities. We show how to use the estimator to drive (goal-oriented) adaptive mesh refinement for diverse Poisson problems and for mixed approximations of the nearly-incompressible elasticity problems. We provide comparisons with various other used estimators. Two open source implementations in the DOLFIN and DOLFINx solvers of the FEniCS Project are provided as supplementary material. [ABSTRACT FROM AUTHOR]

In this paper, based on the KKT system (the first order optimality condition) for elliptic Dirichlet boundary control problems governed by the Laplacian operator, we develop a new variational setting which involves only a fourth order formulation for the adjoint state by cancelling the control and state variables, and prove that the new variational formulation is equivalent to the KKT system, and analyze its well-posedness (unique solvability and stability). Based on the new variational setting, we present a unified framework for corresponding finite element approximation. This idea may be extended to control constraints. In addition, we present residual-based a posteriori error estimators, which are proved to be reliable and efficient. Numerical experiments are reported to validate our theory. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new strategy for solving highly oscillatory two-dimensional Vlasov-Poisson systems by means of a specific version of the parareal algorithm. The novelty consists in using reduced models, obtained from the two-scale convergence theory, for the coarse solving. The reduced models are useful to approximate the original Vlasov-Poisson model at a low computational cost since they are free of high oscillations. Both models are numerically solved in a particle-in-cell framework. We illustrate this strategy with numerical experiments based on long time simulations of a charged beam in a focusing channel and under the influence of a rapidly oscillating external electric field. On the basis of computing times, we provide an analysis of the efficiency of the parareal algorithm in terms of speedup. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a two-grid discretization method for the two-dimensional time-dependent nonlinear Schrödinger equation by mixed finite element methods. Firstly, we solve original nonlinear Schrödinger equation on a much coarser grid. Then, we solve linear Schrödinger equation on the fine grid. We also propose the error estimate of the two-grid solution with the exact solution in L 2 -norm with order O (h k + 1 + H 2 k + 2). It is shown that our two-grid algorithm can achieve asymptotically optimal approximations as long as the mesh sizes satisfy h = O (H 2). Finally, two numerical experiments in the RT 0 space are provided to partly verify the accuracy and efficiency of the two-grid algorithm. [ABSTRACT FROM AUTHOR]

In this paper we present a finite element analysis for a Dirichlet boundary control problem governed by the Stokes equation. The Dirichlet control is considered in a convex closed subset of the energy space H 1 (Ω). Most of the previous works on the Stokes Dirichlet boundary control problem deals with either tangential control or the case where the flux of the control is zero. This choice of the control is very particular and their choice of the formulation leads to the control with limited regularity. To overcome this difficulty, we introduce the Stokes problem with outflow condition and the control acts on the Dirichlet boundary only hence our control is more general and it has both the tangential and normal components. We prove well-posedness and discuss on the regularity of the control problem. The first-order optimality condition for the control leads to a Signorini problem. We develop a two-level finite element discretization by using P 1 elements (on the fine mesh) for the velocity and the control variable and P 0 elements (on the coarse mesh) for the pressure variable. The standard energy error analysis gives 1 2 + δ 2 order of convergence when the control is in H 3 2 + δ (Ω) space. Here we have improved it to 1 2 + δ , which is optimal. Also, when the control lies in less regular space we derived optimal order of convergence up to the regularity. The theoretical results are corroborated by a variety of numerical tests. [ABSTRACT FROM AUTHOR]

Neutron diffusion kinetics equation contains a diffusion type equation and a set of ordinary differential equations, which form a coupled system. In this paper, we give a positivity theorem for neutron diffusion kinetics equation with delayed neutrons and propose a nonlinear positivity-preserving finite volume scheme on general star-shaped quadrilateral meshes, which can deal with discontinuous and anisotropy diffusion coefficient problems. The existence of solution for the nonlinear system is proved under certain constraint on time step size, which is independent of spatial mesh size. Three nonlinear iteration strategies are introduced to solve the nonlinear system. Numerical experiments are performed to demonstrate that the scheme is positivity-preserving and keeps approximately second order accuracy in space. The computational costs of three nonlinear iteration strategies are compared. Another example is also present to show some constraint on time step size is necessary for the convergence of the nonlinear iteration. [ABSTRACT FROM AUTHOR]

The presented article is based on the detection and resolution algorithms that are assigned to the adaptive finite element methods for solid mechanics problems, in which the boundary layer phenomenon cannot be neglected. In the considered problems, the quantity of interest is the global strain energy. The algorithms were applied within the exemplary hpq -adaptive finite element method composed of 3D-based hierarchical modelling and approximation, residual equilibration method of error estimation and a three step adaptive procedure. This procedure is extended through the inclusion of the additional step of adaptation where the boundary layer is a posteriori detected and resolved. The detection algorithm is based on a simple sensitivity analysis performed locally for the finite elements adjacent to the boundary. The detection algorithm is also extended onto: assessment of the boundary layer significance, determination of the optimal ratio of the exponential refinement, and iterative repetition of the refinement. The resolution of the boundary layer was based on the exponential (or graded) refinement of the mesh in the direction normal to the boundary. In the paper, our theoretical considerations, algorithms, and numerical examples are consistently limited to elliptic shells and plates, however the proposed approach can be extended to parabolic and hyperbolic shells. [ABSTRACT FROM AUTHOR]

The presented paper concentrates on the boundary element method (BEM) for the heat equation in three spatial dimensions. In particular, we deal with tensor product space-time meshes allowing for quadrature schemes analytic in time and numerical in space. The spatial integrals can be treated by standard BEM techniques known from three dimensional stationary problems. The contribution of the paper is twofold. First, we provide temporal antiderivatives of the heat kernel necessary for the assembly of BEM matrices and the evaluation of the representation formula. Secondly, the presented approach has been implemented in a publicly available library besthea allowing researchers to reuse the formulae and BEM routines straightaway. The results are validated by numerical experiments in an HPC environment. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse problem for time-fractional advection-dispersion equation in a multi-layer composite medium. The main goal of our paper is to approximate the initial information, which is inaccessible for measurement, from the observation data at a certain point in second layer by constructing a regularized solution using a filter regularization method. Under appropriate regularity assumptions of the exact solution, we give convergence rate of the error between the reconstructed solution and the exact one. A numerical example is provided to illustrate the main results. [ABSTRACT FROM AUTHOR]

Let Γ be a smooth curve inside a two-dimensional rectangular region Ω. In this paper, we consider the Poisson interface problem − ∇ 2 u = f in Ω ∖ Γ with Dirichlet boundary condition such that f is smooth in Ω ∖ Γ and the jump functions [ u ] and [ ∇ u ⋅ n → ] across Γ are smooth along Γ. This Poisson interface problem includes the weak solution of − ∇ 2 u = f + g δ Γ in Ω as a special case. Because the source term f is possibly discontinuous across the interface curve Γ and contains a delta function singularity along the curve Γ, both the solution u of the Poisson interface problem and its flux ∇ u ⋅ n → are often discontinuous across the interface. To solve the Poisson interface problem with singular sources, in this paper we propose a sixth order compact finite difference scheme on uniform Cartesian grids. Our proposed compact finite difference scheme with explicitly given stencils extends the immersed interface method (IIM) to the highest possible accuracy order six for compact finite difference schemes on uniform Cartesian grids, but without the need to change coordinates into the local coordinates as in most papers on IIM in the literature. Also in contrast with most published papers on IIM, we explicitly provide the formulas for all involved stencils and therefore, our proposed scheme can be easily implemented and is of interest to practitioners dealing with Poisson interface problems. Note that the curve Γ splits Ω into two disjoint subregions Ω + and Ω −. The coefficient matrix A in the resulting linear system A x = b , following from the proposed scheme, is independent of any source term f , jump condition g δ Γ , interface curve Γ and Dirichlet boundary conditions, while only b depends on these factors and is explicitly given, according to the configuration of the nine stencil points in Ω + or Ω −. The constant coefficient matrix A facilitates the parallel implementation of the algorithm in case of a large size matrix and only requires the update of the right hand side vector b for different Poisson interface problems. Due to the flexibility and explicitness of the proposed scheme, it can be generalized to obtain the highest order compact finite difference scheme for non-uniform grids as well. We prove the order 6 convergence for the proposed scheme using the discrete maximum principle. Our numerical experiments confirm the sixth accuracy order of the proposed compact finite difference scheme on uniform meshes for the Poisson interface problems with various singular sources. [ABSTRACT FROM AUTHOR]

In this paper, we consider an optimal control problem governed by linear parabolic differential equations with memory. Under the assumption that the corresponding linear parabolic differential equation without memory term is approximately controllable, it is shown that the set of approximate controls is nonempty. The problem is first viewed as a constrained optimal control problem, and then it is approximated by an unconstrained problem with a suitable penalty function. The optimal pair of the constrained problem is obtained as the limit of the optimal pair sequence of the unconstrained problem. The result is proved by using the theory of strongly continuous semigroups and the Banach fixed point theorem. The approximation theorems, which guarantee the convergence of the numerical scheme to the optimal pair sequence, are also proved. Finally, we present an algorithm to compute an optimal control with a numerical example. [ABSTRACT FROM AUTHOR]

Simultaneous image denoising and deblurring is a challenging issue because noise and edges are both high-frequency signals, and eliminating noise while enhancing edges counteracts each other. In this paper, we deal with this difficulty by designing a novel coupled time-fractional diffusion along with a fractional shock filter and developing a powerful discretization approach. The discretization process is based on the finite difference and L 1 approximations for integer and fractional order derivatives, respectively. This scheme not only removes noise and sharpens edges efficiently, but also weakens the staircase effect. The stability analysis of the proposed scheme is also investigated. The experimental results show that the proposed approach outperforms other related approaches and can be applied to large-scale and chest X-Ray images. [ABSTRACT FROM AUTHOR]

Fourth-order differential equations play an important role in many applications in science and engineering. In this paper, we present a three-field mixed finite-element formulation for fourth-order problems, with a focus on the effective treatment of the different boundary conditions that arise naturally in a variational formulation. Our formulation is based on introducing the gradient of the solution as an explicit variable, constrained using a Lagrange multiplier. The essential boundary conditions are enforced weakly, using Nitsche's method where required. As a result, the problem is rewritten as a saddle-point system, requiring analysis of the resulting finite-element discretization and the construction of optimal linear solvers. Here, we discuss the analysis of the well-posedness and accuracy of the finite-element formulation. Moreover, we develop monolithic multigrid solvers for the resulting linear systems. Two and three-dimensional numerical results are presented to demonstrate the accuracy of the discretization and efficiency of the multigrid solvers proposed. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse problem of identifying a pair of function { u (x , t) , r (t) } in the time fractional diffusion equation from two kinds of observations. By virtue of the high-precision Legendre spectral collocation and mollification method in the spatial and time direction severally, we present an efficient algorithm to solve the inverse problem. Finally, two numerical examples explicate the effectiveness and stability of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, a new finite difference unequal-sized Hermite weighted essentially non-oscillatory (US-HWENO) scheme is proposed for solving multi-dimensional Navier-Stokes equations on structured meshes, which could achieve sixth-order accuracy in one dimension and fifth-order accuracy in two and three dimensions, respectively. The high-order spatial reconstruction procedure uses a convex combination of one quintic polynomial and two linear polynomials defined on three unequal-sized central or biased spatial stencils. Compared with the classical finite difference, finite volume, or mixed finite difference and finite volume HWENO schemes [32,33] , the new scheme uses same largest stencil and only three stencils in spatial reconstruction procedures, and the first-order derivative values in auxiliary equations can be obtained directly rather than using complex HWENO-type reconstruction methodologies [32,33]. Moreover, the associated viscosity terms could be directly computed from the function values and first-order derivative values by using the characteristic of the WENO-type spatial reconstructions. Meanwhile, the linear weights can be any positive numbers on condition that their summation equals one, which avoids the problem of negative linear weights in classical HWENO schemes [24,25]. So the new finite difference US-HWENO scheme is more suitable for solving Navier-Stokes equations, and can be easily implemented to multi-dimensions on structured meshes. Some benchmark viscous examples are illustrated to verify the good performance of this new finite difference US-HWENO scheme. [ABSTRACT FROM AUTHOR]

In this paper, a first-order Euler semi-implicit finite element scheme is proposed for solving the two-dimensional incompressible magnetohydrodynamics flows with variable density numerically. The proposed FEM algorithm is unconditionally stable at the full discrete level, which is a key issue in designing efficient algorithms for the multi-physical field problems. When the time step size τ and the mesh size h both are sufficiently small, optimal spatial error estimates in L 2 -norm are presented for the density, velocity and magnetic field by using energy techniques based on uniform regularities of solutions to the time-discrete scheme and the method of mathematical induction. Finally, numerical results are given to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper aims to consider the energy-preserving finite element method (FEM) for the general nonlinear Schrödinger equation with wave operator. Optimal error estimates and superconvergence of the energy-preserving scheme are proved rigorously without any time-step restriction, which cover a general nonlinearity. There are two ingredients in our analysis. First, a temporal-spatial error splitting argument is employed to split the error into temporal error and spatial error. Second, by use of the cut-off function technique for the nonlinearity, the discrete auxiliary problems are introduced so as to verify the well-posedness of temporal semi-discrete and fully-discrete schemes, and to derive the temporal error with order O (τ 2) in H 2 -norm and spatial error with order O (τ 2 + h 2) in H 1 -norm, respectively. In addition, by virtue of the idea of combination between interpolation and projection, as well as the interpolated postprocessing technique, the superclose and superconvergence results of order O (τ 2 + h 2) in H 1 -norm are deduced for the fully-discrete scheme. At last, two numerical examples are provided to confirm theoretical analysis. Here, h is the subdivision parameter, and τ is the time step. [ABSTRACT FROM AUTHOR]

In this paper we propose a mesh-free method for parabolic nonlocal problem. The proposed method comprises web-splines, Euler scheme and Newton's method. We give existence and uniqueness results for the weak formulation at continuous level and prove a priori error bounds for the fully-discrete solution. Finally, we show some numerical results based on web-spline method to confirm our theoretical findings. [ABSTRACT FROM AUTHOR]

Characterizing physical properties of faults, such as their transmissibility, is crucial for performing predictive numerical simulation of subsurface flows, such as those encountered in petroleum engineering and remediation of subsurface contamination. This paper provides a complete investigation of the inverse problem for fault transmissibility in subsurface flow models, under appropriate assumptions on fault structure. In particular, the following aspects are considered: 1) fault modeling and well-posedness of the forward problem; 2) finite element (FEM) discretizations of the forward problem and their rigorous a priori convergence analysis; 3) Well-posedness of the Bayesian inverse problem, FEM discretization of the infinite dimensional Bayesian inverse formulation, and its rigorous a priori analysis. Moreover, computation of the maximum a posteriori (MAP) point via fast inexact Newton-conjugate gradient optimization and a Laplace approximation of the Bayesian posterior are also presented. Numerical results illustrate the use of the proposed fault model in forward and inverse problems for subsurface flows in two dimensional domains with multiple faults. • A priori error analysis of finite element methods for a faulted subsurface flow model. • Well-posedness of the Bayesian inverse problem for fault transmissibility. • FEM discretization of the infinite dimensional Bayesian inverse problem with analysis. • Maximum a posteriori computation via a fast inexact Newton method. • Numerical results illustrating forward and inverse problems of fault models. [ABSTRACT FROM AUTHOR]

This paper is devoted to the numerical analysis of a control constrained distributed optimal control problem subject to a time fractional diffusion equation with non-smooth initial data. The solutions of state and co-state are decomposed into singular and regular parts, and some growth estimates are obtained for the singular parts. By following the variational discretization concept, a full discretization is applied to the corresponding state and co-state equations by using linear conforming finite element method in space and piecewise constant discontinuous Galerkin method in time. Error estimates are derived by employing the growth estimates. In particular, graded temporal grids are adopted to obtain the first-order temporal accuracy. Finally, numerical experiments are performed to verify the theoretical results. [ABSTRACT FROM AUTHOR]

We investigate the error of the (semidiscrete) Galerkin method applied to a semilinear subdiffusion equation in the presence of nonsmooth initial data. The diffusion coefficient is allowed to depend on time. We use the energy method to find optimal bounds on the error under weak natural assumptions on the diffusivity. First, we prove the result for the linear problem and then use the "frozen nonlinearity" technique coupled with various generalizations of Grönwall inequality to carry the result to the semilinear case. The paper ends with numerical illustrations supporting the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we study the numerical approximation of the time fractional Cahn-Hilliard equation with initial singularity. A nonlinear fully discrete scheme is constructed using the L2- 1 σ formula of time fractional order derivative on graded meshes and the Fourier spectral method in space. We present a priori estimate and give the existence and uniqueness of the approximate solution. The stability and convergence of the proposed scheme are proved rigorously. A numerical example is carried out to illustrate the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, a multilevel Monte Carlo ensemble hybridizable discontinuous Galerkin (MLMCE-HDG) method is proposed to solve the convection diffusion equation with random diffusion coefficients. This method combines the MLMC method in probability space, an ensemble approach in time space and the HDG method in physical space. The proposed method possesses a second order accuracy in time space and optimal L 2 convergence rate in physical space. The stability analysis, error estimates and computational cost are provided. Finally, to illustrate the theoretical analyses, several numerical experiments are performed. [ABSTRACT FROM AUTHOR]

In this paper, we propose a decoupled, unconditionally energy stable and charge-conservative finite element method for the inductionless magnetohydrodynamic equations. The time marching is combined with a first order semi-implicit Euler scheme, a first order perturbation term and some delicate implicit-explicit treatments for coupling terms. The finite element discretization is based on mixed conforming elements, where the hydrodynamic unknowns are approximated by stable finite element pairs, and the current density and electric potential are discretized by divergence-conforming face elements and discontinuous volume elements. This fully discrete scheme only needs to solve a linear subsystem at each time step and yields an exactly divergence-free current density directly. We show that the proposed scheme is well-posed and unconditionally energy stable. Under a weak regularity hypothesis on the exact solution, we present the error estimates for velocity, current density and electric potential. Finally, the validity, reliability, and accuracy of the proposed algorithm are supported by several numerical experiments. [ABSTRACT FROM AUTHOR]

• The INTERNODES method is applied to problems in contact mechanics. • We propose a highly efficient preconditioner to solve the sequence of linear systems. • An algorithm is suggested to cheaply solve linear systems with the preconditioning matrix. • The quality of the preconditioner is assessed on small to medium size 2D problems. The mortar finite element method is a well-established method for the numerical solution of partial differential equations on domains displaying non-conforming interfaces. The method is known for its application in computational contact mechanics. However, its implementation remains challenging as it relies on geometrical projections and unconventional quadrature rules. The INTERNODES (INTERpolation for NOn-conforming DEcompositionS) method, instead, could overcome the implementation difficulties thanks to flexible interpolation techniques. Moreover, it was shown to be at least as accurate as the mortar method making it a very promising alternative for solving problems in contact mechanics. Unfortunately, in such situations the method requires solving a sequence of ill-conditioned linear systems. In this paper, preconditioning techniques are designed and implemented for the efficient solution of those linear systems. [ABSTRACT FROM AUTHOR]

This paper investigates consensus control problem for a class of distributed parameter type multi-agent differential inclusion systems with state time-delay by utilizing iterative learning control (ILC). Unlike most ILC literature of nonlinear distributed parameter systems that require an identical virtual leader, the virtual leader is iteratively varying in this work and can be described by any follower agent. Both P -type closed-loop and D -type open-loop ILC are proposed for the distributed parameter type multi-agent differential inclusion systems, which enable the output trajectories of each agent to achieve consensus in the sense of W 1 , 2 norm as the iteration number increases. Meanwhile, consensus control theorems for these protocols are established by using Steiner-type selector, Green's formulas and some technical inequalities. Finally, numerical simulations testify the validity of the proposed methods. [ABSTRACT FROM AUTHOR]

In this paper we propose and analyze a new mixed finite element method for a stationary magneto-hydrodynamic (MHD) model. The method is based on the utilization of a new dual-mixed formulation recently introduced for the Navier-Stokes problem, which is coupled with a classical primal formulation for the Maxwell equations. The latter implies that the velocity and a pseudostress tensor relating the velocity gradient with the convective term for the hydrodynamic equations, together with the magnetic field and a Lagrange multiplier related with the divergence-free property of the magnetic field, become the main unknowns of the system. Then the associated Galerkin scheme can be defined by employing Raviart–Thomas elements of degree k for the aforementioned pseudostress tensor, discontinuous piecewise polynomial elements of degree k for the velocity, Nédélec elements of degree k for the magnetic field and Lagrange elements of degree k for the associated Lagrange multiplier. The analysis of the continuous and discrete problems are carried out by means of the Lax–Milgram lemma, the Banach–Nečas–Babuška and Banach fixed-point theorems, under a sufficiently small data assumption. In particular, the analysis of the discrete scheme requires a quasi-uniformity assumption on mesh. We also develop an a priori error analysis and show that the proposed finite element method is optimal convergent. Finally, some numerical results illustrating the good performance of the method are provided. [ABSTRACT FROM AUTHOR]

In this paper, we establish a novel fully discrete semi-implicit stabilized finite element method for the Cahn-Hilliard-Navier-Stokes phase-field model by using the lowest equal-order (P 1 / P 1 / P 1 / P 1) finite element pair, which consists of the stabilized finite element method based on multiscale enrichment for the spatial discretization and the first order semi-implicit scheme combined with convex splitting approximation for the temporal discretization. We prove that the fully discrete scheme is unconditional energy stable and mass conservative. We also carry out optimal error estimates both in time and space for the phase function, chemical potential and velocity in the appropriate norms. Finally, several numerical experiments are presented to confirm the theoretical results and the efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, a new strategy to establish less time-consuming upwind compact difference method with adjusted dissipation is introduced for solving the incompressible Navier-Stokes (N-S) equations in the streamfunction-velocity form efficiently. By weighted combination of the numerical solutions calculated using the upwind term and the downwind term of the general third-order upwind compact scheme (UCD3), a new fourth-order compact formulation and a third-order upwind compact formulation with adjusted dissipation nature are proposed for computing the first derivatives. Further, they are used to approximate the biharmonic term and the convective terms in the streamfunction-velocity formulation of the N-S equations, respectively. Meanwhile, the first derivatives of the streamfunction (velocities) in the coefficients in the convective terms are solved by the newly proposed fourth-order compact formulation. Temporal discretization for the streamfunction-velocity formulation is addressed with the help of the second-order Crank-Nicolson scheme. Moreover, the newly proposed scheme for the linear models is proved to be unconditionally stable by virtue of the discrete Fourier analysis. Finally, five numerical problems, viz. the analytic solution, Taylor-Green vortex problem, doubly periodic double shear layer flow problem, lid-driven square cavity flow problem and two-sided square cavity flow problem are solved numerically to verify the efficiency and accuracy of the present method. Results solved by the present method match well with the analytic solutions and the existing results proving the accuracy of it. What is more, it is less time-consuming and has lower dissipation than the existing method [20]. [ABSTRACT FROM AUTHOR]

In this paper, we shall present a weak virtual element method for the standard three field poroelasticity problem on polytopal meshes. The flux velocity and pressure are approximated by the low order H (div) virtual element and the piecewise constant, while the elastic displacement is discretized by the H (div) virtual element with some tangential polynomials on element boundaries. A fully discrete scheme is then given by choosing the backward Euler for the time discretization. With some assumptions on the exact solutions, we prove that the convergence order is of order 1 with respect to the meshsize and the time step, and the hidden constants are independent of the parameters of the problems. Some numerical experiments are given to verify the results. [ABSTRACT FROM AUTHOR]

In this paper we seek to recover the mass density of a rectangular taut membrane from the first eigenvalues of the small transverse vibration under supported boundary conditions, provided the mass is a small perturbation of the constant density and is symmetric with respect to both midlines. The proposed method allows for the determination of the generalized Fourier coefficients of the mass density change evaluated on a suitable basis of functions, which naturally arises from the linearized Taylor expansion of the eigenvalues. The reconstruction is based on a sequence of successive linearizations of the inverse problem in a neighborhood of the uniform membrane, and numerical results indicate improvements over other techniques available in the literature. The method has been tested also on mass densities that fall outside the range of small perturbations of the constant density, showing some potential for the reconstruction. [ABSTRACT FROM AUTHOR]

This paper examines a time fractional version of the 2D Fokker-Planck equation involved with the Atangana-Baleanu-Caputo fractional derivative, under the Dirichlet boundary conditions. A fully discretization approach based on the meshless local Petrov-Galerkin method and (3 − β) -order approximation is proposed for this equation. More precisely, we apply the meshless local Petrov-Galerkin method based on the Moving Kriging interpolation to discretize the space domain, and utilize the (3 − β) -order approximation together with the θ -weighted finite difference method to discretize the temporal domain. By implementing this method, we get a solution for the problem by solving a system of algebraic equations. The validity of the method is investigated by solving four examples. [ABSTRACT FROM AUTHOR]

This paper presents novel finite element solvers for Stokes flow that are pressure-robust due to the use of a lifting operator. Specifically, weak Galerkin (WG) finite element schemes are developed for the Stokes problem on quadrilateral and hexahedral meshes. Local Arbogast-Correa or Arbogast-Tao spaces are utilized for construction of discrete weak gradients. The lifting operator lifts WG test functions into H (div) -subspaces and removes pressure dependence of velocity errors. The pressure robustness of these solvers is validated theoretically and illustrated numerically. Comparison with the non-robust classical Taylor-Hood (Q 2 , Q 1) solver is presented. [ABSTRACT FROM AUTHOR]

The dimension-wise splitting with selective relaxation (DSSR) is an efficient iterative method for solving linear systems arising from the discretization of incompressible Navier-Stokes equations. The unconditional convergence properties and optimal relaxation parameters are analyzed at the continuous level on model problems in Gander et al. (2016) [15] In this paper, we provide an algebraically spectral analysis of the DSSR preconditioner when applied to Krylov subspace methods. The analysis indicates that most eigenvalues of the preconditioned matrix are 1, and the corresponding eigenspace is nondeficient, which is an important favorable property for fast convergence of Krylov subspace iterative method. Finally, the preconditioner is explained from the deflation point of view, which provides a clue for developing hybrid direct iterative solvers. [ABSTRACT FROM AUTHOR]

In this paper, we construct an efficient and conservative compact difference scheme based on the scalar auxiliary variable (SAV) approach for Boussinesq Paradigm (BP) equation. The compact difference scheme preserves the mass and discrete modified energy. We prove uniquely solvability of the compact difference scheme and analyze the bounded estimates of the numerical solution. The rates of convergence of second-order in temporal direction and fourth-order in spatial direction are given by using the discrete energy method in detail. Some numerical experiments are given to verify our theoretical analysis. [ABSTRACT FROM AUTHOR]

This paper develops a model for describing a stochastic age-dependent population system (SADPS) with Lévy noise in a polluted environment. As the model includes a nonlinear drift term and Lévy noise, it is difficult, if not impossible, to derive an analytical solution of the model. This study thus aims at developing a numerical algorithm that is based on a modified truncated Euler-Maruyama (EM) method and positivity preserving. We first study global positivity for the solution of the SADPS model in a polluted environment. Subsequently, we present a semi-discrete Galerkin finite element method in an age variable, and construct a full-discrete numerical approximation using modified truncated EM scheme on time to preserve positivity. Under certain suitable regularity conditions, estimates of convergence error of the full-discrete and a semi-discrete scheme are presented. Finally, a numerical example is given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

Recently, a functional named quasi-area functional was proposed in [14] to approximate the area functional in the Plateau-Bézier problem. The quasi-area functional is constructed by a balanced sum among the quasi-harmonic functional, Dirichlet functional and a functional which measures isothermality. It improves greatly the approximation efficiency of existing methods. Therefore in this paper, based on the quasi-area functional, we introduce two methods to study the triangular Plateau-Bézier problem. On the one hand, we minimize directly the functional among all the triangular Bézier surfaces with prescribed border determined by the exterior control points. On the other hand, we compute and analyze the Euler-Lagrange equation of the quasi-area functional. Both methods are illustrated by some representative examples to show the effectiveness. [ABSTRACT FROM AUTHOR]

In this paper, we report numerical simulations for dynamic fracture and fragmentation problems in brittle materials using a recently developed split strain energy diffusive mass-field damage model in terms of standard finite element method (FEM). The enhanced constitutive laws for mass source and mass flux by means of strain energy decomposition are adopted to overcome certain drawbacks of the original non-split model. We present theoretical formulations in a more general way that is suitable and valid for both small and finite deformation regimes. The coupled system of equations is derived in which the momentum is governed to describe the time-dependent deformation of brittle solids whereas the mass-diffusion balance equations are for the evolution of mass density. We also develop simple techniques for determining crack velocity and estimating amount of mass loss. Numerical examples are considered for both small and finite deformations. The accuracy and performance of the developed theory for dynamic brittle fracture are illustrated through comparison and verification, which are to compare the computed results with respect to experimental data and other numerical methods in terms of crack path trajectories, dynamic response, energy dissipation, and other relevant numerical aspects. • Dynamic description of localized mass-field damage model at finite deformation for brittle fracture is presented. • Localized mass-field damage model enhanced by energy decomposition for dynamic analysis is used. • Dynamic crack propagation for problems with complex geometries and fragmentation simulation are studied. • Prediction of dissipated energy agrees well with the theoretical value. • It provides an alternative approach for modeling dynamic fracture in solids. [ABSTRACT FROM AUTHOR]

In this paper, we present a numerical method for the phase-field model of anisotropic ice crystal growth on a spherical surface. The mathematical model includes terms related to the anisotropic interfacial energy, which is defined by the interface angle with respect to a reference angle. One of the natural numerical methods on curved surfaces is a computational technique based on a triangular mesh for the surface in a three-dimensional space. However, it is difficult to compute terms with the interface angle on a triangular mesh. To resolve this problem, we solve the governing equation in Cartesian coordinates after rotating each vertex and the 1-ring neighborhood of the vertex on the triangular mesh. After rotation and interpolation, we numerically solve the phase-field model using a standard finite difference method. We present the results of several tests to demonstrate that the proposed algorithm can recover anisotropic ice crystal growth on a spherical surface. [ABSTRACT FROM AUTHOR]

The time-fractional Burgers-Huxley (TFBH) equation is a typical fractional parabolic equation, it is an evolutionary model describing the propagation of neural pulses. The high-accuracy numerical method for studying TFBH equation has important scientific significance and engineering application value. In this paper, a high-order compact difference scheme is constructed for inhomogeneous TFBH equation. The time-fractional derivative is discretized by L 1 formula, and the spatial derivative is approximated by fourth-order precision compact approximation. We analyzed the existence and uniqueness of difference scheme solution, and proved the stability and convergence of the fourth-order compact scheme using the energy method. Numerical experiments show that the scheme converges to O (τ 2 − α + h 4) under the strong regularity assumption. Under the condition of weak regularity, the scheme converges to O (τ α + h 4). It shows that the scheme has good robustness and high accuracy for solving inhomogeneous TFBH equation. [ABSTRACT FROM AUTHOR]

In this paper, we first formulate a hydrodynamically coupled phase-field model for triblock copolymer melts and then develop a time-marching scheme for it. The numerical scheme is based on a combination of the projection method and the IEQ (Invariant Energy Quadratization) method to achieve linear and second-order accuracy in time. To further simplify its implementation, we introduce a new nonlocal variable and its time evolution equation, so we end up with a fully decoupled scheme. Under the premise of satisfying strict unconditional energy stability, the scheme only needs to solve several completely independent elliptic equations with constant coefficients at each time step. We provide the rigorous energy stability proof, the detailed implementation of the proposed scheme, and further perform some numerical simulations of triblock polymer materials under applied shear flow in 2D and 3D to demonstrate the effectiveness of the developed numerical scheme. [ABSTRACT FROM AUTHOR]

The paper concerns with accurate evaluation of local fields in problems, like those of fracture mechanics, for which extreme, rather than mean values, are of prime significance. We focus on large-scale problems, solved iteratively by a boundary element method (BEM) with employing the kernel independent fast multipole method (KI-FMM) for decreasing the complexity of matrix-to-vector multiplications. We aim to study the accuracy of individual translations far-fields for the major types of kernels (weakly singular, singular and hypersingular) entering the boundary integral equations (BIE) of harmonic and elasticity problems in 2D and 3D. It is revealed, analytically and numerically, that (i) the accuracy of individual translations of fields, generated by sources involving the normal at a source point, is notably less than that by sources independent on the normal; (ii) the accuracy is notably less for a hypersingular BIE, then for a singular BIE; and (iii) under the same complexity, the accuracy of translations, performed by using smooth (circular in 2D, spherical in 3D) equivalent surfaces (ES), is notably better than that when using non-smooth (square in 2D, cubic in 3D) ES. These conclusions yield recommendations for minimizing the complexity without loss of the accuracy of translated fields. [ABSTRACT FROM AUTHOR]

In this paper, an efficient space-time backward substitution method (STBSM) is presented for solving nonlinear transient heat conduction problems in 2D and 3D functionally graded materials (FGMs). To improve the computation efficiency of the traditional backward substitution method (BSM) for transient problems, the space-time radial basis function (STRBF) and space-time trigonometric basis function (STTBF) are introduced for interpolation in the proposed method. The time dimension is regarded as a pseudo-space dimension to generate the space-time Euler metric r T in STRBF, which can be applied to transform the d -dimensional transient governing equation into (d + 1) -dimensional steady-state governing equation. Benefiting from merits of the traditional BSM, an accurate numerical solution is available with only a few of computational points. Several examples are solved to verify the accuracy and efficiency of the proposed method and its merits and flaws are discussed in the end. • A space-time method is attempted for 2D and 3D heat conduction problems in FGMs. • Space-time domain and space-time radial basis function are introduced into backward substitution method. • By introducing pseudo-space-time dimension to transform the origin transient problem into steady-state problem. • The proposed method has excellent computational efficiency. [ABSTRACT FROM AUTHOR]

Transient nonlinear problems play an important role in many engineering problems. Phase-field equations, including the well-known Allen-Cahn and Cahn-Hilliard equations, fall in this category, and have applications in cutting-edge technologies such as modeling the diffusion of lithium (Li) ions in two-phase electrode particles of Li-ion batteries. In this paper, a local meshless method for solving this category of partial differential equations (PDEs) is proposed. The Newton-Kantorovich scheme is employed to transform the nonlinear PDEs to an iterative series of linear ones which can be solved with the proposed method. The accuracy and performance of the method are examined in various linear and nonlinear problems, such as Laplace equation, three dimensional elasticity as well as some abstract mathematical equations with linear or nonlinear boundary conditions. The main focus of the work is on applying the proposed method in solution of the phase-field equations, including the Allen-Cahn and Cahn-Hilliard equations. In addition to homogeneous Neumann boundary condition which has been widely examined in the literature, we also employ a practical nonlinear, inhomogeneous Neumann boundary condition formulation specialized for modeling the diffusion of lithium ions in electrode particles of Li-ion batteries. The generalized- α method is used for time integration of diffusion-type equations to overcome the intrinsic stiffness of the phase-field equations. It is shown that the method is capable of capturing the main features of the phase-field models i.e. phase separation, coarsening and energy decay in closed systems. [ABSTRACT FROM AUTHOR]

This paper proposes, analyzes and tests a second-order time-accurate partitioned method for the coupling dual-porosity-Stokes equations. The algorithm combines a backward differentiation formula and the explicit second order's extrapolation method, and decouples the coupled problem to three sub-domain equations (matrix pressure equation, microfracture pressure equation and Stokes equation) at each time step. To improve the accuracy of the approximate solutions and enhance conservation of mass, the algorithm adds a grad-div stabilization term to the partitioned code and extends it to modular grad-div algorithm. We derive the time stability of the algorithm without the time step restriction and error estimate for fully discretized schemes using finite element spatial discretization. Numerical experiments validate the accuracy and advantage of grad-div stabilization algorithms and support the theoretical analysis. [ABSTRACT FROM AUTHOR]

The WENO-AIM proposed by Vevek et al. [22] can achieve the optimal order for the critical point, and the complex structure of the mapping function of WENO-AIM will substantially increase the computational cost. To reduce the computational cost, we recommended a novel finite-difference mapped WENO method of a rational adaptive operator in this paper. This mapping-type WENO method has a simple framework of achieving the optimal order for critical points of high efficiency. Numerical experiments with one- and two-dimensional benchmark problems of the new mapped method compared with WENO-AIM and other WENO methods. It reveals that the new mapped WENO scheme can further to improve the resolution of WENO-AIM and suppress the numerical oscillation near the discontinuities with higher extra computational efficiency. WENO-AM remains stable with a large CFL number, while WENO-AIM generates non-physical oscillatory. [ABSTRACT FROM AUTHOR]