Energy-minimizing, symmetric finite differences for anisotropic meshes and energy functional extrapolation

Martin Joachim Kühn (1, 2) , Carola Kruse (2) , Ulrich Rüde (3, 2)

1 German Aerospace Center (DLR), Institute for Software Technology, Department for HighPerformance Computing, Cologne, Germany
2 CERFACS - Centre Européen de Recherche et de Formation Avancée en Calcul Scientifique
3 Friedrich-Alexander Universität Erlangen-Nürnberg, Chair for Computer Science 10 - System simulation

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 was 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.

Mots clés: partial differential equation; energy functional; symmetry; anisotropy; extrapola- tion; finite differences; finite elements

Domaines: Analyse numérique [math.NA]

Martin Joachim Kühn, Carola Kruse, Ulrich Rüde. Energy-minimizing, symmetric finite differences for anisotropic meshes and energy functional extrapolation. 2020. ⟨hal-02941899⟩