A simplified primal-dual weak Galerkin finite element method for Fokker-Planck type equations

TL;DR

This paper introduces a simplified primal-dual weak Galerkin finite element method for Fokker-Planck equations, reducing degrees of freedom and improving stability, with proven optimal error estimates and validated numerical results.

Contribution

The paper develops a simplified S-PDWG method with fewer degrees of freedom and better condition number for Fokker-Planck equations, enhancing computational efficiency.

Findings

Fewer degrees of freedom compared to previous methods

Smaller condition number due to new stabilizer

Optimal error estimates in L2 norm

Abstract

A simplified primal-dual weak Galerkin (S-PDWG) finite element method is designed for the Fokker-Planck type equation with non-smooth diffusion tensor and drift vector. The discrete system resulting from S-PDWG method has significantly fewer degrees of freedom compared with the one resulting from the PDWG method proposed by Wang-Wang \cite{WW-fp-2018}. Furthermore, the condition number of the S-PDWG method is smaller than the PDWG method \cite{WW-fp-2018} due to the introduction of a new stabilizer, which provides a potential for designing fast algorithms. Optimal order error estimates for the S-PDWG approximation are established in the $L^2$ norm. A series of numerical results are demonstrated to validate the effectiveness of the S-PDWG method.