Polytopal Meshes Research Articles

Weak Galerkin method for the Biot’s consolidation model

We develop a weak Galerkin (WG) finite element method for the Biot’s consolidation model in the classical displacement–pressure two-field formulation. Weak Galerkin linear finite elements are used for both displacement and pressure approximations in spatial discretizations. Backward Euler scheme is used for temporal discretization in order to obtain an implicit fully discretized scheme. We study the well-posedness of the linear system at each time step and also derive the overall optimal-order convergence of the WG formulation. Such WG scheme is designed on general shape regular polytopal meshes and provides stable and oscillation-free approximation for the pressure without special treatment. Numerical experiments are presented to demonstrate the efficiency and accuracy of the proposed weak Galerkin finite element method.

Anhp-Hybrid High-Order Method for Variable Diffusion on General Meshes

AbstractIn this work, we introduce and analyze anhp-hybrid high-order (hp-HHO) method for a variable diffusion problem. The proposed method is valid in arbitrary space dimension and for fairly general polytopal meshes. Variable approximation degrees are also supported. We provehp-convergence estimates for both the energy- andL2L^{2}-norms of the error, which are the first of this kind for Hybrid High-Order methods. These results hinge on a novelhp-approximation lemma valid for general polytopal elements in arbitrary space dimension. The estimates are additionally fully robust with respect to the heterogeneity of the diffusion coefficient, and show only a mild dependence on the square root of the local anisotropy, improving previous results for HHO methods. The expected exponential convergence behavior is numerically demonstrated on a variety of meshes for both isotropic and strongly anisotropic diffusion problems.

A Hybrid High-Order method for Leray–Lions elliptic equations on general meshes

In this work, we develop and analyze a Hybrid High-Order (HHO) method for steady non-linear Leray–Lions problems. The proposed method has several assets, including the support for arbitrary approximation orders and general polytopal meshes. This is achieved by combining two key ingredients devised at the local level: a gradient reconstruction and a high-order stabilization term that generalizes the one originally introduced in the linear case. The convergence analysis is carried out using a compactness technique. Extending this technique to HHO methods has prompted us to develop a set of discrete functional analysis tools whose interest goes beyond the specific problem and method addressed in this work: (direct and) reverse Lebesgue and Sobolev embeddings for local polynomial spaces, $L^p$-stability and $W^{s,p}$-approximation properties for $L^2$-projectors on such spaces, and Sobolev embeddings for hybrid polynomial spaces. Numerical tests are presented to validate the theoretical results for the original method and variants thereof.

Minimal degree $H(\mathrm {curl})$ and $H(\mathrm {div})$ conforming finite elements on polytopal meshes

Minimal degree $H(\mathrm {curl})$ and $H(\mathrm {div})$ conforming finite elements on polytopal meshes

The mimetic finite difference method for the Landau–Lifshitz equation

The Landau–Lifshitz equation describes the dynamics of the magnetization inside ferromagnetic materials. This equation is highly nonlinear and has a non-convex constraint (the magnitude of the magnetization is constant) which poses interesting challenges in developing numerical methods. We develop and analyze explicit and implicit mimetic finite difference schemes for this equation. These schemes work on general polytopal meshes which provide enormous flexibility to model magnetic devices with various shapes. A projection on the unit sphere is used to preserve the magnitude of the magnetization. We also provide a proof that shows the exchange energy is decreasing in certain conditions. The developed schemes are tested on general meshes that include distorted and randomized meshes. The numerical experiments include a test proposed by the National Institute of Standard and Technology and a test showing formation of domain wall structures in a thin film.

A new weak Galerkin finite element method for elliptic interface problems

A new weak Galerkin (WG) finite element method is introduced and analyzed in this paper for solving second order elliptic equations with discontinuous coefficients and interfaces. Comparing with the existing WG algorithm for solving the same type problems, the present WG method has a simpler variational formulation and fewer unknowns. Moreover, the new WG algorithm allows the use of finite element partitions consisting of general polytopal meshes and can be easily generalized to high orders. Optimal order error estimates in both H1 and L2 norms are established for the present WG finite element solutions. Extensive numerical experiments have been conducted to examine the accuracy, flexibility, and robustness of the proposed WG interface approach. In solving regular elliptic interface problems, high order convergences are numerically confirmed by using piecewise polynomial basis functions of high degrees. Moreover, the WG method is shown to be able to accommodate very complicated interfaces, due to its flexibility in choosing finite element partitions. Finally, in dealing with challenging problems with low regularities, the piecewise linear WG method is capable of delivering a second order of accuracy in L∞ norm for both C1 and H2 continuous solutions.

Gradient schemes: Generic tools for the numerical analysis of diffusion equations

The gradient scheme framework is based on a small number of properties and encompasses a large number of numerical methods for diffusion models. We recall these properties and develop some new generic tools associated with the gradient scheme framework. These tools enable us to prove that classical schemes are indeed gradient schemes, and allow us to perform a complete and generic study of the well-known (but rarely well-studied) mass lumping process. They also allow an easy check of the mathematical properties of new schemes, by developing a generic process for eliminating unknowns via barycentric condensation, and by designing a concept of discrete functional analysis toolbox for schemes based on polytopal meshes.

Spectral properties and conservation laws in Mimetic Finite Difference methods for PDEs

The Mimetic Finite Difference (MFD) methods for PDEs mimic crucial properties of mathematical systems: duality and self-adjointness of differential operators, conservation laws and properties of the solution on general polytopal meshes. In this article the structure and the spectral properties of the linear systems derived by the spatial discretization of diffusion problem are analysed. In addition, the numerical approximation of parabolic equations is discussed where the MFD approach is used in the space discretization while implicit ϑ-method and explicit Runge–Kutta–Chebyshev schemes are used in time discretization. Moreover, we will show how the numerical solution preserves certain conservation laws of the theoretical solution.

Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes

AbstractA new weak Galerkin (WG) finite element method is introduced and analyzed in this article for the biharmonic equation in its primary form. This method is highly robust and flexible in the element construction by using discontinuous piecewise polynomials on general finite element partitions consisting of polygons or polyhedra of arbitrary shape. The resulting WG finite element formulation is symmetric, positive definite, and parameter‐free. Optimal order error estimates in a discrete H2 norm is established for the corresponding WG finite element solutions. Error estimates in the usual L2 norm are also derived, yielding a suboptimal order of convergence for the lowest order element and an optimal order of convergence for all high order of elements. Numerical results are presented to confirm the theory of convergence under suitable regularity assumptions. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1003–1029, 2014

Carbamylation of normal and uremic erythrocyte membrane protein

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.

Extension of a boundary representation technique for the description of N dimensional polytopes

A polytope is a geometrical figure bounded by portions of lines, planes, or hyperplanes; e.g., in two dimensions it is a polygon, in three dimensions a polyhedron. Boundary representation is an approved and accepted method for representing three dimensional objects, particularly for discrete surface polyhedra. It produces an efficient and effective conceptual model, typically stored within a data structure known as a polygonal mesh. This representation technique is particularly suited for further extension, resulting in the development of a polytopal mesh for the methodical description of N dimensional polytopes.

On the construction and display of a polytopal mesh for the N-dimensional hypercube and ( N + 1) simplex

By considering the analytical properties of polytopes it is possible to design a general data structure, a polytopal mesh, to represent such N-dimensional objects. Further investigation of the N-dimensional hypercube leads to the construction of a ternary code for representing the hypercube and the interconnection between levels of the polytopal mesh. This technique has been extended to other convex polytopes and specific examples of the N-dimensional cross polytope and simplex are given. Various projection techniques are used to display the convex polytopes on a two dimensional viewport.