• Abstract

In this paper, we introduce the weak Galerkin (WG) method for solving the coupled Stokes and Darcy-Forchheimer flows problem with the Beavers-Joseph-Saffman interface condition in bounded domains. We define the WG spaces in the polygonal meshes and construct corresponding discrete schemes. We prove the existence and uniqueness of the WG scheme by the discrete inf-sup condition and monotone operator theory. Then, we derive the optimal error estimates for the velocity and pressure. Numerical experiments are presented to verify the efficiency of the WG method.

• General Summary

• Abstract

Superconvergence of differential structure on discretized surfaces is studied in this paper. The newly introduced geometric supercloseness provides us with a fundamental tool to prove the superconvergence of gradient recovery on deviated surfaces. An algorithmic framework for gradient recovery without exact geometric information is introduced. Several numerical examples are documented to validate the theoretical results.

• General Summary

• Abstract

This paper is concerned with the new error analysis of a Hodge-decomposition based finite element method for the time-dependent Ginzburg-Landau equations in superconductivity. In this approach, the original equation of magnetic potential $\boldsymbol{A}$ is replaced by a new system consisting of four scalar variables. As a result, the conventional Lagrange finite element method (FEM) can be applied to problems defined on non-smooth domains. It is known that due to the low regularity of $\boldsymbol{A},$ conventional FEM, if applied to the original Ginzburg-Landau system directly, may converge to the unphysical solution. The main purpose of this paper is to establish an optimal error estimate for the order parameter in spatial direction, as previous analysis only gave a sub-optimal convergence rate analysis for all three variables due to coupling of variables. The analysis is based on a nonstandard quasi-projection for ψ and the corresponding negative-norm estimate for the classical Ritz projection. Our numerical experiments confirm the optimal convergence of $ψ_h.$

• General Summary