Optimal Order Error Estimates Research Articles

Virtual element method for solving a viscoelastic contact problem with long memory

In this paper, we use the virtual element method to solve a history-dependent hemivariational inequality arising in contact problems. The contact problem concerns the deformation of a viscoelastic body with long memory, subjected to a contact condition with non-monotone normal compliance and unilateral constraints. A fully discrete scheme based on the trapezoidal rule for the discretization of the time integration and the virtual element method for the spatial discretization are analyzed. We provide a unified priori error analysis for both internal and external approximations. For the linear virtual element method, we obtain the optimal order error estimate. Finally, three numerical examples are reported, providing numerical evidence of the theoretically predicted optimal convergence orders.

Analytical and numerical studies of a cancer invasion model with nonlocal diffusion

Abstract This article investigates the implicit Euler–Galerkin finite element method applied to a cancer invasion model with nonlocal diffusion. This model describes the normal cell density, tumor cell density, excess H + ion concentration, extracellular matrix, and extracellular matrix active metalloproteinases. The primary goal of this work is to utilize numerical solutions to comprehend cancer invasion and transmission within the patient. The Faedo–Galerkin approximation method is initially used to establish the existence of a weak solution, followed by deriving the a priori error estimates of optimal order. Additionally, the implicit Euler–Galerkin finite element method is employed as a numerical tool for solving the given cancer invasion parabolic system. Furthermore, a priori error bounds and convergence estimates for the fully discrete problem are derived. Finally, numerical simulations are conducted to validate the theoretical conclusions and enhance the understanding of how cancer spreads within the patient.

Low regularity error analysis for an H(div)-conforming discontinuous Galerkin approximation of Stokes problem

In this paper, we derive an improved error estimate for the H(div)-conforming discontinuous Galerkin (DG) approximation of the Stokes equations, assuming only minimal regularity on the exact solution. The estimate relies on both a priori and a posteriori analysis, and thus is called a medius error analysis. More precisely, we proved an optimal order error estimate under the assumption (u,p)∈H1+s(Ω)×Hs(Ω) with any s∈(0,1]. Extension to the standard interior penalty DG methods is also explored. Finally, numerical results are provided to verify our theoretical findings.

Error estimates and numerical simulations of a thermoviscoelastic contact problem with damage and long memory

This paper aims to investigate a thermal frictional contact model with damage and long memory effects. We consider a deformable body made of viscoelastic material and assume the process to be dynamic. The material is expected to adhere to the Kelvin–Voigt constitutive law, with damage and thermal effects incorporated. The variational formulation of the model results in a coupled system comprising a history-dependent hemivariational inequality governing the displacement field, a parabolic variational inequality describing the damage field and an evolution equation for the temperature field. In the analysis of this system, we initially introduce a fully discrete scheme, and then concentrate on deriving error estimates of numerical solutions. An optimal order error estimate is attained under some appropriate solution regularity assumptions. At the tail of this manuscript, numerical simulations are provided for the contact problem to validate the theoretical results.

A global space–time Trefftz DG scheme for the time-dependent isotropic elastic wave equations

In this paper we are concerned with Trefftz discretizations of the time-dependent linear elastic wave equation in arbitrary space dimensional domains Ω⊂Rd(d∈N). We propose a Trefftz discontinuous Galerkin method, construct Trefftz basis functions and prove that the corresponding approximate solutions possess optimal-order error estimates with respect to the meshwidth. A space–time domain decomposition preconditioner is introduced for the system generated by the proposed variational formula. Besides, we propose the global Trefftz DG method combined with local DG methods to solve the nonhomogeneous model. The numerical results verify the validity of the theoretical results, and show that the resulting approximate solutions possess high accuracy, and the proposed preconditioner is effective.

Weak Galerkin methods for elliptic interface problems on curved polygonal partitions

This paper presents a new weak Galerkin (WG) method for elliptic interface problems on general curved polygonal partitions. The method’s key innovation lies in its ability to transform the complex interface jump condition into a more manageable Dirichlet boundary condition, simplifying the theoretical analysis significantly. The numerical scheme is designed by using locally constructed weak gradient on the curved polygonal partitions. We establish error estimates of optimal order for the numerical approximation in both discrete H1 and L2 norms. Additionally, we present various numerical results that serve to illustrate the robust numerical performance of the proposed WG interface method.

Numerical analysis of a history-dependent mixed hemivariational-variational inequality in contact problems

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.

An H-div finite element method for the Stokes equations on polytopal meshes

In this paper, we introduce an H(div)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H({\ ext {div}})$$\\end{document} finite element method on polygonal and polyhedral meshes for solving the Stokes equations in the primary velocity–pressure formulation. An H(div)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H({\ ext {div}})$$\\end{document} finite element on polygons or polyhedra is introduced to approximate the velocity so that the method is pressure robust and produces exact divergence free solutions, when the discontinuous Pk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_k$$\\end{document} finite element is adopted for the pressure approximation. In addition, this method is stabilizer-free, in handling the H1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1$$\\end{document} non-conformity. Optimal order error estimates are established for the method. Numerical tests are conducted to confirm the theory.

Mathematical and numerical analysis for PDE systems modeling intravascular drug release from arterial stents and transport in arterial tissue.

This paper is concerned with the PDE (partial differential equation) and numerical analysis of a modified one-dimensional intravascular stent model. It is proved that the modified model has a unique weak solution by using the Galerkin method combined with a compactness argument. A semi-discrete finite-element method and a fully discrete scheme using the Euler time-stepping have been formulated for the PDE model. Optimal order error estimates in the energy norm are proved for both schemes. Numerical results are presented, along with comparisons between different decoupling strategies and time-stepping schemes. Lastly, extensions of the model and its PDE and numerical analysis results to the two-dimensional case are also briefly discussed.

On discrete ground states of rotating Bose–Einstein condensates

The ground states of Bose–Einstein condensates in a rotating frame can be described as constrained minimizers of the Gross–Pitaevskii energy functional with an angular momentum term. In this paper we consider the corresponding discrete minimization problem in Lagrange finite element spaces of arbitrary polynomial order and we investigate the approximation properties of discrete ground states. In particular, we prove a priori error estimates of optimal order in the L 2 L^2 - and H 1 H^1 -norm, as well as for the ground state energy and the corresponding chemical potential. A central issue in the analysis of the problem is the missing uniqueness of ground states, which is mainly caused by the invariance of the energy functional under complex phase shifts. Our error analysis is therefore based on an Euler–Lagrange functional that we restrict to certain tangent spaces in which we have local uniqueness of ground states. This gives rise to an error decomposition that is ultimately used to derive the desired a priori error estimates. We also present numerical experiments to illustrate various aspects of the problem structure.

An upwind-mixed volume element method on changing meshes for compressible miscible displacement problem

Numerical simulation of oil-water miscible displacement is discussed in this paper, and a compressible problem of energy mathematics is solved potentially. The mathematical model defined by a nonlinear system includes mainly two partial differential equations (PDEs): a parabolic equation for the pressure and a convection-diffusion equation for the saturation. The pressure is obtained by a conservative mixed finite volume element method (MFVE). The computational accuracy is improved for Darcy velocity. A conservative upwind mixed finite volume element method (UMFVE) is applied to compute the saturation on changing meshes. The diffusion is discretized by a mixed finite volume element method, and the convection is computed by upwind differences. The upwind method can solve convection-dominated diffusion equations accurately and avoids numerical dispersion and nonphysical oscillation. The saturation and the adjoint vector function are obtained simultaneously. An optimal order error estimates is obtained. Finally, numerical examples are provided to show the accuracy, efficiency and possible applications.

A space-time Petrov-Galerkin method for the two-dimensional regularized long-wave equation

In this work, a space-time Petrov-Galerkin (STPG) method is used to numerically analyze the two-dimensional regularized long-wave (RLW) equation. The STPG method is a nonstandard finite element method, that is, both of the spatial and temporal variables of this method are discretized by finite element method. Therefore, it can display the superiority of the finite element method in both spatial and temporal directions. In particular, it is easy to obtain space-time high accuracy and is unconditionally stable, thus this method is very appropriate for solving the nonlinear convection-diffusion equations. We prove the existence and uniqueness of the approximate solution and derive an optimal order error estimate in the maximum norm without requiring any compatibility between the space and time mesh size. In addition, some numerical experiments are presented to validate the accuracy and efficiency of the proposed method. Also, we provide several numerical experiments to confirm the conservation properties of discrete mass, energy and momentum for both the single and double-soliton waves.

High order Morley elements for biharmonic equations on polytopal partitions

This paper introduces an extension of the Morley element for approximating solutions to biharmonic equations. Traditionally limited to piecewise quadratic polynomials on triangular elements, the extension leverages weak Galerkin finite element methods to accommodate higher degrees of polynomials and the flexibility of general polytopal elements. By utilizing the Schur complement of the weak Galerkin method, the extension allows for fewest local degrees of freedom while maintaining sufficient accuracy and stability for the numerical solutions. The numerical scheme incorporates locally constructed weak tangential derivatives and weak second order partial derivatives, resulting in an accurate approximation of the biharmonic equation. Optimal order error estimates in both a discrete H2 norm and the usual L2 norm are established to assess the accuracy of the numerical approximation. Additionally, numerical results are presented to validate the developed theory and demonstrate the effectiveness of the proposed extension.

Optimal order error estimates of a mixed discontinuous Galerkin method with BDF-2 scheme for transient Darcy flow

In this paper, we study the numerical solution of Darcy flow via a mixed discontinuous Galerkin method. The two-step Euler backward formula is used for the time variable discretization and a mixed discontinuous Galerkin method is applied to spatial variable discretization. The stability results are established for our mixed discontinuous Galerkin numerical scheme in a fully discrete framework. Optimal error estimates for our unconditional scheme are derived for both the velocity and pressure variables. Finally, a numerical experiment is provided to approve the accuracy and efficiency of the proposed numerical method.

Numerical solutions for Biharmonic interface problems via weak Galerkin finite element methods

This paper focuses on the advancement of weak Galerkin (WG) finite element methods for addressing two-dimensional and three-dimensional Biharmonic interface problems with polygonal/polyhedral meshes. The WG method has been demonstrated to be accurate and efficient, providing optimal order error estimates in discrete H2 and standard L2 norms. A series of extensive numerical tests are conducted to validate the WG solutions, showcasing the flexibility, stability, and robustness of the proposed method for handling both smooth and complicated interfaces.

Curved elements in weak Galerkin finite element methods

A weak Galerkin finite element method is devised for the Poisson equation with Dirichlet boundary value when curved elements are employed in the numerical scheme. Optimal order error estimates are derived for the weak Galerkin solution in both the H1-norm and the L2-norm. Numerical results are produced to demonstrate the performance of the weak Galerkin method on general curved partitions.

A conforming discontinuous Galerkin finite element method for Brinkman equations

In this paper, we present a conforming discontinuous Galerkin (CDG) finite element method for Brinkman equations. The velocity stabilizer is removed by employing the higher degree polynomials to compute the weak gradient. The theoretical analysis shows that the CDG method is actually stable and accurate for the Brinkman equations. Optimal order error estimates are established in H1 and L2 norm. Finally, numerical experiments verify the stability and accuracy of the CDG numerical scheme.

An H1-Galerkin Space-Time Mixed Finite Element Method for Semilinear Convection–Diffusion–Reaction Equations

In this paper, the semilinear convection–diffusion–reaction equation is split into a lower-order system by introducing the auxiliary variable q=a(x)ux. An H1-Galerkin space-time mixed finite element method for the lower-order system is then constructed. The proposed method applies the finite element method to discretize the time and space directions simultaneously and does not require checking the Ladyzhenskaya–Babusˇka–Brezzi (LBB) compatibility constraints, which differs from the traditional mixed finite element method. The uniqueness of the approximate solutions u and q are proven. The L2(L2) norm optimal order error estimates of the approximate solution u and q are derived by introducing the space-time projection operator. The numerical experiment is presented to verify the theoretical results. Furthermore, by comparing with the classical H1-Galerkin mixed finite element scheme, the proposed scheme can easily improve computational accuracy and time convergence order by changing the basis function.

Error estimates of high-order compact finite difference schemes for the nonlinear abcd Boussinesq systems

Abstract In this paper, some fourth-order compact finite difference schemes are derived and analyzed for the nonlinear $abcd$ Boussinesq systems. The optimal order error estimates for the semidiscrete compact finite difference schemes with different cases of dispersion coefficients $a,\ b,\ c,\ d$, are presented. The third-order and fourth-order linearized implicit multistep schemes are adopted for time discretization, and numerical experiments are conducted on the model problems. Numerical results show that the proposed schemes have high accuracy and are consistent with the theoretical analysis.

A weak Galerkin finite element method for 1D semiconductor device simulation models

In this paper, we study a weak Galerkin (WG) finite element method for semiconductor device simulations. We consider the one-dimensional drift–diffusion (DD) and high-field (HF) models, which involves not only first derivative convection terms but also second derivative diffusion terms, as well as a coupled Poisson potential equation. The main difficulties in the analysis include the treatment of the nonlinearity and coupling of the models. The weak Galerkin finite element method adopts piecewise polynomials of degree k for the approximations of electron concentration and electric potential in the interior of elements, and piecewise polynomials of degree k+1 for the discrete weak derivative space. The optimal order error estimates in a discrete H1 norm and the standard L2 norm are derived. Numerical experiments are presented to illustrate our theoretical analysis. Moreover, numerical schemes also work out for the discontinuous diffusion coefficient problems.