Lennard Kamenski

A New Anisotropic Mesh Adaptation Method Based upon Hierarchical A Posteriori Error Estimates

A new anisotropic mesh adaptation strategy for finite element solution of elliptic differential equations is presented. It generates anisotropic adaptive meshes as quasi-uniform ones in some metric space, with the metric tensor being computed based on hierarchical a posteriori error estimates. A global hierarchical error estimate is employed in this study to obtain reliable directional information of the solution. Instead of solving the global error problem exactly, which is costly in general, we solve it iteratively using the symmetric Gaus–Seidel method. Numerical results show that a few GS iterations are sufficient for obtaining a reasonably good approximation to the error for use in anisotropic mesh adaptation. The new method is compared with several strategies using local error estimators or recovered Hessians. Numerical results are presented for a selection of test examples and a mathematical model for heat conduction in a thermal battery with large orthotropic jumps in the material coefficients.

A geometric discretization and a simple implementation for variational mesh generation and adaptation

We present a simple direct discretization for functionals used in the variational mesh generation and adaptation. Meshing functionals are discretized on simplicial meshes and the Jacobian matrix of the continuous coordinate transformation is approximated by the Jacobian matrices of affine mappings between elements. The advantage of this direct geometric discretization is that it preserves the basic geometric structure of the continuous functional, which is useful in preventing strong decoupling or loss of integral constraints satisfied by the functional. Moreover, the discretized functional is a function of the coordinates of mesh vertices and its derivatives have a simple analytical form, which allows a simple implementation of variational mesh generation and adaptation on computer. Since the variational mesh adaptation is the base for a number of adaptive moving mesh and mesh smoothing methods, the result in this work can be used to develop simple implementations of those methods. Numerical examples are given.

CONDITIONING OF FINITE ELEMENT EQUATIONS WITH ARBITRARY ANISOTROPIC MESHES

This is the published version, also available here: http://dx.doi.org/10.1090/S0025-5718-2014-02822-6. First published in Math. Comput. in 2014, published by the American Mathematical Society

How a Nonconvergent Recovered Hessian Works in Mesh Adaptation

Hessian recovery has been commonly used in mesh adaptation for obtaining the required magnitude and direction information of the solution error. Unfortunately, a recovered Hessian from a linear finite element approximation is nonconvergent in general as the mesh is refined. It has been observed numerically that adaptive meshes based on such a nonconvergent recovered Hessian can nevertheless lead to an optimal error in the finite element approximation. This also explains why Hessian recovery is still widely used despite its nonconvergence. In this paper we develop an error bound for the linear finite element solution of a general boundary value problem under a mild assumption on the closeness of the recovered Hessian to the exact one. Numerical results show that this closeness assumption is satisfied by the recovered Hessian obtained with commonly used Hessian recovery methods. Moreover, it is shown that the finite element error changes gradually with the closeness of the recovered Hessian. This provides a...

A Comparative Numerical Study of Meshing Functionals for Variational Mesh Adaptation

We present a comparative numerical study for three functionals used for variational mesh adaptation. One of them is a generalisation of Winslows variable diffusion functional while the others are based on equidistribution and alignment. These functionals are known to have nice theoretical properties and work well for most mesh adaptation problems either as a stand-alone variational method or combined within the moving mesh framework. Their performance is investigated numerically in terms of equidistribution and alignment mesh quality measures. Numerical results in 2D and 3D are presented.

STABILITY OF EXPLICIT ONE-STEP METHODS FOR P1-FINITE ELEMENT APPROXIMATION OF LINEAR DIFFUSION EQUATIONS ON ANISOTROPIC MESHES

We study the stability of explicit one-step integration schemes for the linear finite element approximation of linear parabolic equations. The derived bound on the largest permissible time step is tight for any mesh and any diffusion matrix within a factor of

A study on using hierarchical basis error estimates in anisotropic mesh adaptation for the finite element method

2(d+1)

Stability of explicit Runge--Kutta methods for high order finite element approximation of linear parabolic equations

, where

Adaptive Finite Elements with Anisotropic Meshes

d

Anisotropic Mesh Adaptation for Variational Problems Using Error Estimation Based on Hierarchical Bases

is the spatial dimension. Both full mass matrix and mass lumping are considered. The bound reveals that the stability condition is affected by two factors. The first depends on the number of mesh elements and corresponds to the classic bound for the Laplace operator on a uniform mesh. The second factor reflects the effects of the interplay of the mesh geometry and the diffusion matrix. It is shown that it is not the mesh geometry itself but the mesh geometry in relation to the diffusion matrix that is crucial to the stability of explicit methods. When the mesh is uniform in the metric specified by the inverse of the diffusion matrix, the stability condition is comparable to the situation with the Laplace operator on a uniform mesh...