Energy-Minimizing, Symmetric Discretizations for Anisotropic Meshes and Energy Functional Extrapolation

Carola Kruse,Martin Joachim Kühn,Ulrich Rüde

doi:10.1137/21m1397520

Abstract

Self-adjoint differential operators often arise from variational calculus on energy functionals. In this case, a direct discretization of the energy functional induces a discretization of the differential operator. Following this approach, the discrete equations are naturally symmetric if the energy functional is self-adjoint, a property that may be lost when using standard difference formulas on nonuniform meshes or when the differential operator has varying coefficients. Low order finite difference or finite element systems can be derived by this approach in a systematic way and on logically structured meshes they become compact difference formulas. Extrapolation formulas used on the discrete energy can then lead to higher oder approximations of the differential operator. A rigorous analysis is presented for extrapolation used in combination with nonstandard integration rules for finite elements. Extrapolation can likewise be applied on matrix-free finite difference stencils. In our applications, both schemes show up to quartic order of convergence.

Highlights

Self-adjoint differential operators are common in many applications
For finite differences, a standard derivation of the discretization can lead to a nonsymmetric system matrix when the grid is irregular or when variable material parameters appear in the partial differential equation or the related energy functional J(u)
We present a natural approach to obtain symmetric finite difference stencils for anisotropic meshes by considering the energy functional corresponding to the partial differential equation similar as proposed in [25]

Summary

Introduction

Self-adjoint differential operators are common in many applications. When discretizing such operators it is often essential to maintain the symmetry. We present a natural approach to obtain symmetric finite difference stencils for anisotropic meshes by considering the energy functional corresponding to the partial differential equation similar as proposed in [25]. We use a nonstandard integration rule from [15, 11] to show equality between an extrapolated stiffness matrix from linear nodal basis functions and the stiffness matrix from a quadratic nodal basis set.

Results

Conclusion