On Convergence of Support Operator Method Schemes for Differential Rotational Operations on Tetrahedral Meshes Applied to Magnetohydrodynamic Problems

Abstract

The problem of constructing and justifying the discrete algorithms of the support operator method for numerical modeling of differential repeated rotational operations of vector analysis (curlcurl) in application to problems of magnetohydrodynamics is considered. Difference schemes of the support operator method on the unstructured meshes do not approximate equations in the local sense. Therefore, it is necessary to prove the convergence of these schemes to the exact solution, which is possible after analyzing the error structure of their approximation. For this analysis, a decomposition of the space of mesh vector functions into an orthogonal direct sum of subspaces of potential and vortex fields is introduced. Generalized centroid-tensor metric representations of repeated operations of tensor analysis (div, grad, and curl) are constructed. Representations have flux-circulation properties that are integrally consistent on spatial meshes of irregular structure. On smooth solutions of the model magnetostatic problem on a tetrahedral mesh with the first order of accuracy in the rms sense, the convergence of the constructed difference schemes is proved. The algorithms constructed in this work can be used to solve physical problems with discontinuous magnetic viscosity, dielectric permittivity, or thermal resistance of the medium.