Superconvergence analysis of the decoupled and linearized mixed finite element methods for unsteady incompressible MHD equations

Abstract

The purpose of this article is to explore the superconvergence behavior of the first-order backward-Euler (BE) implicit/explicit fully discrete schemes for the unsteady incompressible MHD equations with low-order mixed finite element method (MFEM) by utilizing the scalar auxiliary variable (SAV) and zero-energy-contribution (ZEC) methods. Through dealing with linear terms in implicit format and nonlinear terms in explicit format, the original problem is decomposed into several subproblems, which effectively reduces the amount of calculation. Particularly, a new high-precision estimation is given, which acts as a requisite role in getting the expected results. Following this, combined with a simple, effective and economic interpolation post-processing approach, the superclose and superconvergence error estimates of the decoupled and linearized fully discrete finite element SAV-BE scheme are rigorously derived. And the derivation process is also applicable to the ZEC-BE scheme. Finally, the corresponding numerical simulations are carried out to confirm the accuracy and reliability of our theoretical findings.