Polytopal Meshes Research Articles

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.

A posteriori error estimate of the weak Galerkin finite element method solving the Stokes problems on polytopal meshes

AbstractIn this paper, we propose an a posteriori error estimate of the weak Galerkin finite element method (WG‐FEM) solving the Stokes problems with variable coefficients. Its error estimator, based on the property of Stokes' law conservation, Helmholtz decomposition and bubble functions, yields global upper bound and local lower bound for the approximation error of the WG‐FEM. Error analysis is proved to be valid under the mesh assumptions of the WG‐FEM and the way can be extended to other FEMs with the property of Stokes' law conservation, for example, discontinuous Galerkin (DG) FEMs. Finally, we verify the performance of error estimator by performing a few numerical examples.

GTMesh: A Highly Efficient C++ Template Library for Numerical Schemes on General Topology Meshes

This article introduces GTMesh, an open-source C++ library providing data structures and algorithms that facilitate the development of numerical schemes on general polytopal meshes. After discussing the features and limitations of the existing open-source alternatives, we focus on the theoretical description of geometry and the topology of conforming polytopal meshes in an arbitrary-dimensional space, using elements from graph theory. The data structure for mesh representation is explained. The main part of the article focuses on the implementation of data structures and algorithms (computation of measures, centers, normals, cell coloring) by using State-of-the-Art template metaprogramming techniques for maximum performance. The geometrical algorithms are designed to be valid regardless of the dimension of the underlying space. As an integral part of the library, a template implementation of class reflection in C++ has been created, which is sufficiently versatile and suitable for the development of numerical and data I/O algorithms working with generic data types. Finally, the use of GTMesh is demonstrated on a simple example of solving the heat equation by the finite volume method.

A pressure-robust HHO method for the solution of the incompressible Navier–Stokes equations on general meshes

Abstract In a recent work (Castanon Quiroz & Di Pietro (2020) A hybrid high-order method for the incompressible Navier–Stokes problem robust for large irrotational body forces. Comput. Math. Appl., 79, 2655–2677), we have introduced a pressure-robust hybrid high-order method for the numerical solution of the incompressible Navier–Stokes equations on matching simplicial meshes. Pressure-robust methods are characterized by error estimates for the velocity that are fully independent of the pressure. A crucial question was left open in that work, namely whether the proposed construction could be extended to general polytopal meshes. In this paper, we provide a positive answer to this question. Specifically, we introduce a novel divergence-preserving velocity reconstruction that hinges on the solution inside each element of a mixed problem on a subtriangulation, then use it to design discretizations of the body force and convective terms that lead to pressure robustness. An in-depth theoretical study of the properties of this velocity reconstruction, and their reverberation on the scheme, is carried out for arbitrary polynomial degrees $k\geq 0$ and meshes composed of general polytopes. The theoretical convergence estimates and the pressure robustness of the method are confirmed by an extensive panel of numerical examples.

Exact sequences of conforming finite element spaces with interface constraints for macro polytopal meshes

Exact sequences of conforming finite element spaces with interface constraints for macro polytopal meshes

A Conforming DG Method for the Biharmonic Equation on Polytopal Meshes

A Conforming DG Method for the Biharmonic Equation on Polytopal Meshes

A mixed virtual element method for Biot's consolidation model.

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 virtual element and the piecewise constant, while the elastic displacement is discretized by the 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.

Analysis of a weak Galerkin method for second-order elliptic equations with minimal regularity on polytopal meshes

We propose an error analysis of a weak Galerkin method for second-order elliptic equations with minimal regularity requirements, i.e., the exact solution u belongs to H1+s(Ω) with s>0 possibly close to zero. In this case, the conormal derivative of u is not well-defined on the mesh faces, which results in a legitimacy problem of the error equation originally derived under smooth assumption. We employ the weak meaning of the conormal derivative introduced in [Ern, A., & Guermond, J. L. (2021). Quasi-optimal Nonconforming Approximation of Elliptic PDEs with Contrasted Coefficients andH1+s,s>0, Regularity. Foundations of Computational Mathematics, 1–36.] to establish new error equation and optimal a priori error estimates for low regular solutions on polytopal meshes. We also investigate the corresponding elliptic eigenvalue problems. The numerical experiments verify the theoretical results and agree with the benchmark eigenvalue problem.

The MHM Method for Linear Elasticity on Polytopal Meshes

Abstract The multiscale hybrid-mixed (MHM) method consists of a multi-level strategy to approximate the solution of boundary value problems with heterogeneous coefficients. In this context, we propose a new family of finite elements for the linear elasticity equation defined on coarse polytopal partitions of the domain. The finite elements rely on face degrees of freedom associated with multiscale bases obtained from local Neumann problems with piecewise polynomial interpolations on faces. We establish sufficient conditions on the fine-scale interpolations such that the MHM method is well-posed and optimally convergent under local regularity conditions. Also, a multi-level error analysis demonstrates that the MHM method achieves convergence without refining the coarse partition. The upshot is that the Poincaré and Korn’s inequalities do not degenerate, and then the convergence arises on general meshes. Two- and three-dimensional numerical tests assess theoretical results and verify the robustness of the method on a multi-layer media case.

A Hybrid High-Order Method for Quasilinear Elliptic Problems of Nonmonotone Type

In this paper, we design and analyze a Hybrid-High Order (HHO) approximation for a class of quasilinear elliptic problems of nonmonotone type. The proposed method has several advantages, for instance, it supports arbitrary order of approximation and general polytopal meshes. The key ingredients involve local reconstruction and high-order stabilization terms. Existence and uniqueness of the discrete solution are shown by Brouwer's fixed point theorem and contraction result. A priori error estimate is shown in discrete energy norm that shows optimal order convergence rate. Numerical experiments are performed to substantiate the theoretical results.

BDDC Preconditioners for Divergence Free Virtual Element Discretizations of the Stokes Equations

The virtual element method (VEM) is a new family of numerical methods for the approximation of partial differential equations, where the geometry of the polytopal mesh elements can be very general. The aim of this article is to extend the balancing domain decomposition by constraints preconditioner to the solution of the saddle-point linear system arising from a VEM discretization of the two-dimensional Stokes equations. Under suitable hypotesis on the choice of the primal unknowns, the preconditioned linear system results symmetric and positive definite, thus the preconditioned conjugate gradient method can be used for its solution. We provide a theoretical convergence analysis estimating the condition number of the preconditioned linear system. Several numerical experiments validate the theoretical estimates, showing the scalability and quasi-optimality of the method proposed. Moreover, the solver exhibits a robust behavior with respect to the shape of the polygonal mesh elements. We also show that a faster convergence could be achieved with an easy to implement coarse space, slightly larger than the minimal one covered by the theory.

Conforming virtual element approximations of the two-dimensional Stokes problem

The virtual element method (VEM) is a Galerkin approximation method that extends the finite element method to polytopal meshes. In this paper, we present two different conforming virtual element formulations for the numerical approximation of the Stokes problem that work on polygonal meshes. The velocity vector field is approximated in the virtual element spaces of the two formulations, while the pressure variable is approximated through discontinuous polynomials. Both formulations are inf-sup stable and convergent with optimal convergence rates in the L2 and energy norm. We assess the effectiveness of these numerical approximations by investigating their behavior on a representative benchmark problem. The observed convergence rates are in accordance with the theoretical expectations and a weak form of the zero-divergence constraint is satisfied at the machine precision level.

Mixed virtual element method for the Helmholtz transmission eigenvalue problem on polytopal meshes

Abstract In this paper we propose and analyze a mixed virtual element method for the Helmholtz transmission eigenvalue problem, which is a fourth order, nonlinear and non-self-adjoint eigenvalue problem and is important in the inverse scattering theory. The conforming virtual element is used for discretization. We prove correct spectral approximation and error estimates of the discrete scheme. Finally, we show several numerical examples to verify the theoretical results and present the performance of different stabilization parameters and the comparison with the $C^{1}$ virtual element method.

Convergent adaptive hybrid higher-order schemes for convex minimization

This paper proposes two convergent adaptive mesh-refining algorithms for the hybrid high-order method in convex minimization problems with two-sided p-growth. Examples include the p-Laplacian, an optimal design problem in topology optimization, and the convexified double-well problem. The hybrid high-order method utilizes a gradient reconstruction in the space of piecewise Raviart–Thomas finite element functions without stabilization on triangulations into simplices or in the space of piecewise polynomials with stabilization on polytopal meshes. The main results imply the convergence of the energy and, under further convexity properties, of the approximations of the primal resp. dual variable. Numerical experiments illustrate an efficient approximation of singular minimizers and improved convergence rates for higher polynomial degrees. Computer simulations provide striking numerical evidence that an adopted adaptive HHO algorithm can overcome the Lavrentiev gap phenomenon even with empirical higher convergence rates.

A Mixed Finite-Element Method on Polytopal Mesh

In this paper, we introduce new stable mixed finite elements of any order on polytopal mesh for solving second-order elliptic problem. We establish optimal order error estimates for velocity and super convergence for pressure. Numerical experiments are conducted for our mixed elements of different orders on 2D and 3D spaces that confirm the theory.

Development of a LDG method on polytopal mesh with optimal order of convergence

A Local discontinuous Galerkin (LDG) finite element method on triangular mesh was introduced and analyzed in Castillo et al. (2000) for elliptic problem with suboptimal order of convergence for the flux variable. The purpose of this work is to develop a LDG finite element method on polytopal mesh that achieves optimal convergence rate for both unknowns, potential u and its gradient q. In our new LDG method, u and q are approximated by polynomials of degree k and k−1 respectively. Optimal order of convergence are obtained for both unknowns. Numerical results in 2d and 3d are presented to confirm the theory.

Bridging the multiscale hybrid-mixed and multiscale hybrid high-order methods

We establish the equivalence between the Multiscale Hybrid-Mixed (MHM) and the Multiscale Hybrid High-Order (MsHHO) methods for a variable diffusion problem with piecewise polynomial source term. Under the idealized assumption that the local problems defining the multiscale basis functions are exactly solved, we prove that the equivalence holds for general polytopal (coarse) meshes and arbitrary approximation orders. We also leverage the interchange of properties to perform a unified convergence analysis, as well as to improve on both methods.

A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem

<p style='text-indent:20px;'>The Virtual Element Method (VEM) is a Galerkin approximation method that extends the Finite Element Method (FEM) to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element method for the numerical approximation of the Stokes problem to polygonal meshes in the framework of the virtual element method. In particular, we consider a straightforward application of the virtual element approximation space for scalar elliptic problems to the vector case and approximate the pressure variable through discontinuous polynomials. We assess the effectiveness of the numerical approximation by investigating the convergence on a manufactured solution problem and a set of representative polygonal meshes. We numerically show that this formulation is convergent with optimal convergence rates except for the lowest-order case on triangular meshes, where the method coincides with the <inline-formula><tex-math id="M1">\begin{document}$ {\mathbb{P}}_{{1}}-{\mathbb{P}}_{{0}} $\end{document}</tex-math></inline-formula> Scott-Vogelius scheme, and on square meshes, which are situations that are well-known to be unstable.</p>

A reduced-order Weak Galerkin finite element algorithm based on POD technique for parabolic problem on polytopal mesh

Weak Galerkin (WG method) is a numerical method based on the weak functions and weak partial derivatives, which has been widely developed in recent decades. In the implementation, different degrees of freedom (DOFs) are distributed in the element interior and boundary respectively, which leads to a great impact on the computational complexity. In our work, we propose a new reduced-order weak galerkin finite element (RO-WG) method with seldom DOFs, where the proper orthogonal decomposition (POD) technique is adopted to save CPU time. The parabolic equation is considered for a test problem. Some numerical examples are given to prove the superiority of this method, where the CPU time can be reduced a lot.

A priori and a posteriori error estimates for a virtual element method for the non-self-adjoint Steklov eigenvalue problem

Abstract In this paper we analyze a virtual element method (VEM) for the non-self-adjoint Steklov eigenvalue problem. The conforming VEM on polytopal meshes is used for discretization. We analyze the correct spectral approximation of the discrete scheme and prove an a priori error estimate for the discrete eigenvalues and eigenfunctions. The convergence order of a discrete eigenvalue may decrease if the corresponding eigenfunction has a singularity and it can be improved on a locally refined mesh. The VEM has great flexibility in handling computational meshes. These facts motivate us to construct a computable a posteriori error estimator for the VEM and prove its reliability and efficiency. This estimator can be applied to very general polytopal meshes with hanging nodes. Finally, we show numerical examples to verify the theoretical results, including optimal convergence of discrete eigenvalues on uniformly refined meshes of a square domain and a cube domain, and we demonstrate the efficiency of the estimator on adaptively refined meshes on an L-shaped domain and also discuss the influence of stabilization parameters on the virtual element approximation.