We present a new finite volume scheme for anisotropic heterogeneous diffusion problems on unstructured irregular grids, which simultaneously gives an approximation of the solution and of its gradient. In the case of simplicial meshes, the approximate solution is shown to converge to the continuous ones as the size of the mesh tends to 0, and an error estimate is given. In the general case, we propose a slightly modified scheme for which we again prove the convergence, and give an error estimate. An easy implementation method is then proposed, and the efficiency of the scheme is shown on various types of grids.
A simple and efficient interface-fitted mesh generation algorithm is developed in this paper. This algorithm can produce a local anisotropic fitting mixed mesh which consists of both triangles and quadrilaterals near the interface. A new finite element method is proposed for second order elliptic interface problems based on the resulting mesh. Optimal approximation capabilities on anisotropic elements are proved in both the $H^1$ and $L^2$ norms. The discrete system is usually ill-conditioned due to anisotropic and small elements near the interface. Thereupon, a multigrid method is presented t…
We consider spectral mixed discontinuous Galerkin finite element discretizations of the Lamé system of linear elasticity in polyhedral domains in $\mathbb{R}^3$. In order to resolve possible corner, edge, and corner-edge singularities, anisotropic geometric edge meshes consisting of hexahedral elements are applied. We perform a computational study on the discrete inf-sup stability of these methods, and especially focus on the robustness with respect to the Poisson ratio close to the incompressible limit (i.e. the Stokes system). Furthermore, under certain realistic assumptions (for analytic da…
This article considers a new discretization scheme for conservation laws. The discretization setting is based on a discontinuous Galerkin scheme in combination with an approximation space that contains high-order polynomial modes as well as piece-wise constant modes on a sub-grid. The high-order modes can continuously be suppressed with a penalty function that is based on a sensor which is intertwined with the approximation space. Numerical tests finally illustrate the performance of this scheme.
We investigate a mortar technique for mixed finite element approximations of Darcy flow on non-matching grids in which the normal flux is chosen as the coupling variable. It plays the role of a Lagrange multiplier to impose weakly continuity of pressure. In the mixed formulation of the problem, the normal flux is an essential boundary condition and it is incorporated with the use of suitable extension operators. Two such extension operators are considered and we analyze the resulting formulations with respect to stability and convergence. We further generalize the theoretical results, showing …
We construct a new nonlinear finite volume (FV) scheme for highly anisotropic diffusion equations, that satisfies the discrete minimum-maximum principle. The construction relies on the linearized scheme satisfying less restrictive monotonicity conditions than those of an M-matrix, based on a weakly regular matrix splitting and using the Cartesian structure of the mesh (extension to quadrilateral meshes is also possible). The resulting scheme, obtained by expressing fluxes as nonlinear combinations of linear fluxes, has a larger stencil than other nonlinear positivity preserving or minimum-maxi…
A family of conforming mixed finite elements with mass lumping on triangular grids are presented for linear elasticity. The stress field is approximated by symmetric $H({\rm div})-P_k (k\geq 3)$ polynomial tensors enriched with higher order bubbles so as to allow mass lumping, which can be viewed as the Hu-Zhang elements enriched with higher order interior bubble functions. The displacement field is approximated by $C^{-1}-P_{k-1}$ polynomial vectors enriched with higher order terms to ensure the stability condition. For both the proposed mixed elements and their mass lumping schemes, optimal …
For the second lowest-order Raviart--Thomas mixed method, we prove that the canonical interpolant and finite element solution for the vector variable in elliptic problems are superclose in the $H(\text{div})$-norm on mildly structured meshes, where most pairs of adjacent triangles form approximate parallelograms. We then develop a family of postprocessing operators for Raviart--Thomas mixed elements on triangular grids by using the idea of local least squares fittings. Super-approximation property of the postprocessing operators for the lowest and second lowest order Raviart--Thomas elements i…
In this paper, we explore methods for computing wall-normal derivatives used for calculating wall skin friction and heat transfer over a solid wall in unstructured simplex-element (triangular/tetrahedral) grids generated by anisotropic grid adaptation. Simplex-element grids are considered as efficient and suitable for automatic grid generation and adaptation, but present a challenge to accurately predict wall-normal derivatives. For example, wall-normal derivatives computed by a simple finite-difference approximation, as typically done in practical fluid-dynamics simulation codes, are often co…
A mixed finite element method for nearly incompressible elasticity and Stokes equations using primal and dual meshes with quadrilateral and hexahedral grids
We develop a new multipoint stress mixed finite element method for linear elasticity with weakly enforced stress symmetry on simplicial grids. Motivated by the multipoint flux mixed finite element method for Darcy flow, the method utilizes the lowest order Brezzi-Douglas-Marini finite element spaces for the stress and the vertex quadrature rule in order to localize the interaction of degrees of freedom. This allows for local stress elimination around each vertex. We develop two variants of the method. The first uses a piecewise constant rotation and results in a cell-centered system for displa…
This paper introduces a new family of mixed finite elements for solving a mixed formulation of the biharmonic equations in two and three dimensions. The symmetric stress $\bmσ=-\nabla^{2}u$ is sought in the Sobolev space $H({\rm{div}}\bm{div},Ω;\mathbb{S})$ simultaneously with the displacement $u$ in $L^{2}(Ω)$. Stemming from the structure of $H(\bm{div},Ω;\mathbb{S})$ conforming elements for the linear elasticity problems proposed by J. Hu and S. Zhang, the $H({\rm{div}}\bm{div},Ω;\mathbb{S})$ conforming finite element spaces are constructed by imposing the normal continuity of $\bm{div}\bmσ$…
The numerical simulation of physical processes in the underground frequently entails challenges related to the geometry and/or data. The former are mainly due to the shape of sedimentary layers and the presence of fractures and faults, while the latter are connected to the properties of the rock matrix which might vary abruptly in space. The development of approximation schemes has recently focused on the overcoming of such difficulties with the objective of obtaining numerical schemes with good approximation properties. In this work we carry out a numerical study on the performances of the Mi…
We develop a multipoint stress mixed finite element method for linear elasticity with weak stress symmetry on quadrilateral grids, which can be reduced to a symmetric and positive definite cell centered system. The method is developed on simplicial grids in [4]. The method utilizes the lowest order Brezzi-Douglas-Marini finite element spaces for the stress and the trapezoidal quadrature rule in order to localize the interaction of degrees of freedom, which allows for local stress elimination around each vertex. We develop two variants of the method. The first uses a piecewise constant rotation…
Nonlocal multicontinuum (NLMC) upscaling of mixed dimensional coupled flow problem for embedded and discrete fracture models
Texas A&M University（德克萨斯农工大学）；North-Eastern Federal University（东北联邦大学）；The Chinese University of Hong Kong (CUHK)（香港中文大学）；University of Texas at Austin（德克萨斯大学奥斯汀分校）
A strong two-stage explicit/implicit approach combined with mixed finite element methods for a three-dimensional nonlinear radiation-conduction model in anisotropic media
In many applications the accurate representation of the computational domain is a key factor to obtain reliable and effective numerical solutions. Curved interfaces, which might be internal, related to physical data, or portions of the physical boundary, are often met in real applications. However, they are often approximated leading to a geometrical error that might become dominant and deteriorate the quality of the results. Underground problems often involve the motion of fluids where the fundamental governing equation is the Darcy law. High quality velocity fields are of paramount importanc…
We continue the research on the asymptotic and preasymptotic decay of singular numbers for tensor product Hilbert-Sobolev type embeddings in high dimensions with special emphasis on the influence of the underlying dimension $d$. The main focus in this paper lies on tensor products involving univariate Sobolev type spaces with different smoothness. We study the embeddings into $L_2$ and $H^1$. In other words, we investigate the worst-case approximation error measured in $L_2$ and $H^1$ when only $n$ linear samples of the function are available. Recent progress in the field shows that accurate b…
An Efficient Parity-Blocked Method for Band-Structure Computation of 3D Anisotropic Phononic Crystals
Southeast University（东南大学）；Nanjing Normal University（南京师范大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS)（上海数学与交叉学科研究院）；National Yang Ming Chiao Tung University（国立阳明交通大学）
In this paper, we construct a class of Mixed Generalized Multiscale Finite Element Methods for the approximation on a coarse grid for an elliptic problem in thin two-dimensional domains. We consider the elliptic equation with homogeneous boundary conditions on the domain walls. For reference solution of the problem, we use a Mixed Finite Element Method on a fine grid that resolves complex geometry on the grid level. To construct a lower dimensional model, we use the Mixed Generalized Multiscale Finite Element Method, which is based on some multiscale basis functions for velocity fields. The co…
Summation-by-parts (SBP) finite difference methods have several desirable properties for second-order wave equations. They combine the computational efficiency of narrow-stencil finite difference operators with provable stability on curvilinear multiblock grids. While several techniques for boundary and interface conditions exist, weak imposition via simultaneous approximation terms (SATs) is perhaps the most flexible one. Although SBP methods have been applied to elastic wave equations many times, an SBP-SAT method for general anisotropic elastic wave equations has not yet been presented in t…
Mixed-dimensional elliptic equations exhibiting a hierarchical structure are commonly used to model problems with high aspect ratio inclusions, such as flow in fractured porous media. We derive general abstract estimates based on the theory of functional a posteriori error estimates, for which guaranteed upper bounds for the primal and dual variables and two-sided bounds for the primal-dual pair are obtained. We improve on the abstract results obtained with the functional approach by proposing four different ways of estimating the residual errors based on the extent the approximate solution ha…
A Mixed-FEM approximation with uniform conservation of the exponential stability for a class of anisotropic port-Hamiltonian system and its application to LQ control
