Polytopal Meshes Research Articles

A Stabilizer Free Weak Galerkin Method for the Biharmonic Equation on Polytopal Meshes

A new stabilizer free weak Galerkin (WG) method is introduced and analyzed for the biharmonic equation. Stabilizing/penalty terms are often necessary in the finite element formulations with discont...

A stabilizer-free weak Galerkin finite element method on polytopal meshes

A stabilizing/penalty term is often used in finite element methods with discontinuous approximations to enforce connection of discontinuous functions across element boundaries. Removing stabilizers from discontinuous Galerkin finite element methods will simplify formulations and reduce programming complexity significantly. The goal of this paper is to introduce a stabilizer free weak Galerkin (WG) finite element method for second order elliptic equations on polytopal meshes. This new WG method keeps a simple symmetric positive definite form and can work on polygonal/polyhedral meshes. Optimal order error estimates are established for the corresponding WG approximations in both a discrete H1 norm and the L2 norm. Numerical results are presented verifying the theorem.

A hybrid high‐order formulation for a Neumann problem on polytopal meshes

AbstractIn this work, we study a hybrid high‐order (HHO) method for an elliptic diffusion problem with Neumann boundary condition. The proposed method has several features, such as: (a) the support of arbitrary approximation order polynomial at mesh elements and faces on polytopal meshes, (b) the design of a local (element‐wise) potential reconstruction operator and a local stabilization term, that weakly enforces the matching between local element‐ and face‐based on degrees of freedom, and (c) cheap computational cost, thanks to static condensation and compact stencil. We prove the well‐posedness of our HHO formulation, and obtain the optimal error estimates, according to previous study. Implementation aspects are thoroughly discussed. Finally, some numerical examples are provided, which are in agreement with our theoretical results.

A posteriori error estimate for discontinuous Galerkin finite element method on polytopal mesh

AbstractIn this work, we derive a posteriori error estimates for discontinuous Galerkin finite element method on polytopal mesh. We construct a reliable and efficient a posteriori error estimator on general polygonal or polyhedral meshes. An adaptive algorithm based on the error estimator and DG method is proposed to solve a variety of test problems. Numerical experiments are performed to illustrate the effectiveness of the algorithm.

Gradient Recovery for the BEM‐based FEM and VEM

AbstractIn this article, we propose new gradient recovery schemes for the BEM‐based Finite Element Method (BEM‐based FEM) and Virtual Element Method (VEM). Supporting general polytopal meshes, the BEM‐based FEM and VEM are highly flexible and efficient tools for the numerical solution of boundary value problems in two and three dimensions. We construct the recovered gradient from the gradient of the finite element approximation via local averaging. For the BEM‐based FEM, we show that, under certain requirements on the mesh, superconvergence of the recovered gradient is achieved, which means that it converges to the true gradient at a higher rate than the untreated gradient. Moreover, we propose a simple and very efficient a posteriori error estimator, which measures the difference between the unprocessed and recovered gradient as an error indicator. Since the BEM‐based FEM and VEM are specifically suited for adaptive refinement, the resulting adaptive algorithms perform very well in numerical examples.

The Gradient Discretisation Method for Linear Advection Problems

Abstract We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence analysis of various numerical schemes, corresponding to the methods known to be GDMs, such as finite elements (conforming or non-conforming, standard or mass-lumped), finite volumes on rectangular or simplicial grids, and other recent methods developed for general polytopal meshes. The scheme is of centred type, with added linear or non-linear numerical diffusion. We complement the convergence analysis with numerical tests based on the mass-lumped ℙ 1 {\mathbb{P}_{1}} conforming and non-conforming finite element and on the hybrid finite volume method.

A discontinuous Galerkin least-squares method for div–curl systems

A discontinuous Galerkin (DG) least-squares finite element method is developed for solving a div–curl problem. This finite element method leads to a simple symmetric positive definite system and can work on a general polytopal mesh. Optimal convergence rate in the energy norm is established. The numerical results confirm the proposed theory.

Extended virtual element method for the Laplace problem with singularities and discontinuities

In this paper, we propose the extended virtual element method (X-VEM) to treat singularities and crack discontinuities that arise in the Laplace problem. The virtual element method (VEM) is a stabilized Galerkin formulation on arbitrary polytopal meshes, wherein the basis functions are implicit (virtual)—they are not known explicitly nor do they need to be computed within the problem domain. Suitable projection operators are used to decompose the bilinear form on each element into two parts: a consistent term that reproduces the first-order polynomial space and a correction term that ensures stability. A similar approach is pursued in the X-VEM with a few notable extensions. To capture singularities and discontinuities in the discrete space, we augment the standard virtual element space with an additional contribution that consists of the product of virtual nodal basis (partition-of-unity) functions with enrichment functions. For discontinuities, basis functions are discontinuous across the crack and for singularities a weakly singular enrichment function that satisfies the Laplace equation is chosen. For the Laplace problem with a singularity, we devise an extended projector that maps functions that lie in the extended virtual element space onto linear polynomials and the enrichment function, whereas for the discontinuous problem, the consistent element stiffness matrix has a block-structure that is readily computed. An adaptive homogeneous numerical integration method is used to accurately and efficiently (no element-partitioning is required) compute integrals with integrands that are weakly singular. Once the element projection matrix is computed, the same steps as in the standard VEM are followed to compute the element stabilization matrix. Numerical experiments are performed on quadrilateral and polygonal (convex and nonconvex elements) meshes for the problem of an L-shaped domain with a corner singularity and the problem of a cracked membrane under mode III loading, and results are presented that affirm the sound accuracy and demonstrate the optimal rates of convergence in the L2 norm and energy of the proposed method.

Provably positive high-order schemes for ideal magnetohydrodynamics: analysis on general meshes

This paper proposes and analyzes arbitrarily high-order discontinuous Galerkin (DG) and finite volume methods which provably preserve the positivity of density and pressure for the ideal MHD on general meshes. Unified auxiliary theories are built for rigorously analyzing the positivity-preserving (PP) property of MHD schemes with a HLL type flux on polytopal meshes in any space dimension. The main challenges overcome here include establishing relation between the PP property and discrete divergence of magnetic field on general meshes, and estimating proper wave speeds in the HLL flux to ensure the PP property. In 1D case, we prove that the standard DG and finite volume methods with the proposed HLL flux are PP, under condition accessible by a PP limiter. For multidimensional conservative MHD system, standard DG methods with a PP limiter are not PP in general, due to the effect of unavoidable divergence-error. We construct provably PP high-order DG and finite volume schemes by proper discretization of symmetrizable MHD system, with two divergence-controlling techniques: locally divergence-free elements and a penalty term. The former leads to zero divergence within each cell, while the latter controls the divergence error across cell interfaces. Our analysis reveals that a coupling of them is important for positivity preservation, as they exactly contribute the discrete divergence-terms absent in standard DG schemes but crucial for ensuring the PP property. Numerical tests confirm the PP property and the effectiveness of proposed PP schemes. Unlike conservative MHD system, the exact smooth solutions of symmetrizable MHD system are proved to retain the positivity even if the divergence-free condition is not satisfied. Our analysis and findings further the understanding, at both discrete and continuous levels, of the relation between the PP property and the divergence-free constraint.

Anisotropic Error Estimates of the Linear Nonconforming Virtual Element Methods

A refined a priori error analysis of the lowest order (linear) nonconforming Virtual Element Method (VEM) for approximating a model Poisson problem is developed in both 2D and 3D. A set of new geometric assumptions is proposed on shape regularity of polytopal meshes. A new error equation for the lowest order (linear) nonconforming VEM is derived for any choice of stabilization, and a new stabilization using a projection on an extended element patch is introduced for the error analysis on anisotropic elements.

Fast Numerical Integration on Polytopic Meshes with Applications to Discontinuous Galerkin Finite Element Methods

In this paper we present efficient quadrature rules for the numerical approximation of integrals of polynomial functions over general polygonal/polyhedral elements that do not require an explicit construction of a sub-tessellation into triangular/tetrahedral elements. The method is based on successive application of Stokes’ theorem; thereby, the underlying integral may be evaluated using only the values of the integrand and its derivatives at the vertices of the polytopic domain, and hence leads to an exact cubature rule whose quadrature points are the vertices of the polytope. We demonstrate the capabilities of the proposed approach by efficiently computing the stiffness and mass matrices arising from hp-version symmetric interior penalty discontinuous Galerkin discretizations of second-order elliptic partial differential equations.

V-cycle Multigrid Algorithms for Discontinuous Galerkin Methods on Non-nested Polytopic Meshes

In this paper we analyse the convergence properties of V-cycle multigrid algorithms for the numerical solution of the linear system of equations arising from discontinuous Galerkin discretization of second-order elliptic partial differential equations on polytopal meshes. Here, the sequence of spaces that stands at the basis of the multigrid scheme is possibly non nested and is obtained based on employing agglomeration with possible edge/face coarsening. We prove that the method converges uniformly with respect to the granularity of the grid and the polynomial approximation degree p, provided that the number of smoothing steps, which depends on p, is chosen sufficiently large.

Optimal power diagrams via function approximation

In this paper, we present a novel method for generating cell complexes with anisotropy conforming to the Hessian of an arbitrary given function. This is done by variationally optimizing the discontinuous piecewise linear approximation of the given functions over power diagrams. The resulting cell complexes corresponding to the approximations are referred to as Optimal Power Diagram (OPD). A hybrid optimization technique, coupling a modified Monte Carlo method with a local search strategy, is tailored for effectively solving the specific optimization task. In contrast to the Optimal Voronoi Tessellation (OVT) method (Budninskiy et al., 2016), our OPD method does not restrict the target functions to be convex, providing more diverse classes of tessellations of the domain. Furthermore, our OPD method generally yields smaller approximation errors than the OVT method, which uses underlaid approximants. We conduct several experiments to demonstrate the efficacy of our optimization algorithm in finding good local minima and generating high-quality anisotropic polytopal meshes.

A discontinuous least-squares finite-element method for second-order elliptic equations

ABSTRACTIn this paper, a discontinuous least-squares (DLS) finite-element method is introduced. The novelty of this work is twofold, to develop a DLS formulation that works for general polytopal meshes and to provide rigorous error analysis for it. This new method provides accurate approximations for both the primal and the flux variables. We obtain optimal-order error estimates for both the primal and the flux variables. Numerical examples are tested for polynomials up to degree 4 on non-triangular meshes, i.e. on rectangular and hexagonal meshes.

A weak Galerkin least-squares finite element method for div–curl systems

In this paper, we introduce a weak Galerkin least-squares method for solving div–curl problem. This finite element method leads to a symmetric positive definite system and has the flexibility to work with general meshes such as hybrid mesh, polytopal mesh and mesh with hanging nodes. Error estimates of the finite element solution are derived. The numerical examples demonstrate the robustness and flexibility of the proposed method.

Conservative polytopal mimetic discretization of the incompressible Navier–Stokes equations

We discretize the incompressible Navier–Stokes equations on a polytopal mesh by using mimetic reconstruction operators. The resulting method conserves discrete mass, momentum, and kinetic energy in the inviscid limit, and determines the vorticity such that the global vorticity is consistent with the boundary conditions. To do this we introduce a dual mesh and show how the dual mesh can be completed to a cell-complex. We present existing mimetic reconstruction operators in a new symmetric way applicable to arbitrary dimension, use these to interpolate between primal and dual mesh and derive properties of these operators. Finally, we test both 2- and 3-dimensional versions of the method on a variety of complicated meshes to show its wide applicability. We numerically test the convergence of the method and verify the derived conservation statements.

Unified formulation and analysis of mixed and primal discontinuous skeletal methods on polytopal meshes

We propose in this work a unified formulation of mixed and primal discretization methods on polyhedral meshes hinging on globally coupled degrees of freedom that are discontinuous polynomials on the mesh skeleton. To emphasize this feature, these methods are referred to here as discontinuous skeletal. As a starting point, we define two families of discretizations corresponding, respectively, to mixed and primal formulations of discontinuous skeletal methods. Each family is uniquely identified by prescribing three polynomial degrees defining the degrees of freedom, and a stabilization bilinear form which has to satisfy two properties of simple verification: stability and polynomial consistency. Several examples of methods available in the recent literature are shown to belong to either one of those families. We then prove new equivalence results that build a bridge between the two families of methods. Precisely, we show that for any mixed method there exists a corresponding equivalent primal method, and the converse is true provided that the gradients are approximated in suitable spaces. A unified convergence analysis is carried out delivering optimal error estimates in both energy- and L2-norms.

Discontinuous Galerkin gradient discretisations for the approximation of second-order differential operators in divergence form

We include in the gradient discretisation method (GDM) framework two numerical schemes based on discontinuous Galerkin approximations: the symmetric interior penalty Galerkin (SIPG) method, and the scheme obtained by averaging the jumps in the SIPG method. We prove that these schemes meet the main mathematical gradient discretisation properties on any kind of polytopal mesh, by adapting discrete functional analysis properties to our precise geometrical hypotheses. Therefore, these schemes inherit the general convergence properties of the GDM, which hold for instance in the cases of the p-Laplace problem and of the anisotropic and heterogeneous diffusion problem. This is illustrated by simple 1D and 2D numerical examples.

Discontinuous Skeletal Gradient Discretisation methods on polytopal meshes

In this work we develop arbitrary-order Discontinuous Skeletal Gradient Discretisations (DSGD) on general polytopal meshes. Discontinuous Skeletal refers to the fact that the globally coupled unknowns are broken polynomials on the mesh skeleton. The key ingredient is a high-order gradient reconstruction composed of two terms: (i) a consistent contribution obtained mimicking an integration by parts formula inside each element and (ii) a stabilising term for which sufficient design conditions are provided. An example of stabilisation that satisfies the design conditions is proposed based on a local lifting of high-order residuals on a Raviart–Thomas–Nédélec subspace. We prove that the novel DSGDs satisfy coercivity, consistency, limit-conformity, and compactness requirements that ensure convergence for a variety of elliptic and parabolic problems. Links with Hybrid High-Order, non-conforming Mimetic Finite Difference and non-conforming Virtual Element methods are also studied. Numerical examples complete the exposition.

Implementation of Discontinuous Skeletal methods on arbitrary-dimensional, polytopal meshes using generic programming

Discontinuous Skeletal methods approximate the solution of boundary-value problems by attaching discrete unknowns to mesh faces (hence the term skeletal) while allowing these discrete unknowns to be chosen independently on each mesh face (hence the term discontinuous). Cell-based unknowns, which can be eliminated locally by a Schur complement technique (also known as static condensation), are also used in the formulation. Salient examples of high-order Discontinuous Skeletal methods are Hybridizable Discontinuous Galerkin methods and the recently-devised Hybrid High-Order methods. Some major benefits of Discontinuous Skeletal methods are that their construction is dimension-independent and that they offer the possibility to use general meshes with polytopal cells and non-matching interfaces. In this work, we show how this mathematical flexibility can be efficiently replicated in a numerical software using generic programming. We describe a number of generic algorithms and data structures for high-order Discontinuous Skeletal methods within a “write once, run on any kind of mesh” framework. The computational efficiency of the implementation is assessed on the Poisson model problem discretized using various polytopal meshes and the Hybrid High-Order method.