Low Regularity Primal-Dual Weak Galerkin Finite Element Methods for Convection-Diffusion Equations

Abstract

We propose a numerical method for convection-diffusion problems under low regularity assumptions. We derive the method and analyze it using the primal-dual weak Galerkin (PDWG) finite element framework. The Euler-Lagrange formulation resulting from the PDWG scheme yields a system of equations involving not only the equation for the primal variable but also its adjoint for the dual variable. We show that the proposed PDWG method is stable and convergent. We also derive a priori error estimates for the primal variable in the H -norm for ∈ [0, 1 2 ). A series of numerical tests that validate the theory are presented as well.

A New Numerical Method for Div-Curl Systems with Low Regularity Assumptions

“…Two major challenges for the div-curl model problem (1.1)-(1.3) are the low-regularity of the true solution u and the difficulties in approximating the normal ε-harmonic vector space H εn,0 (Ω) due to the topological complexity of the domain Ω. The goal of this paper is to address both challenges through a new numerical method based on the primal-dual weak Galerkin (PDWG) finite element aproach originated in [23] and further developed in [20,25,26,7,22,13] for various model problems. Our PDWG numerical method for (1.1)-(1.3) has two prominent features over the existing numerical methods: (1) it offers an effective approximation of the normal ε-harmonic vector space H εn,0 (Ω) regardless of the topology of the domain Ω; and (2) it provides an accurate and reliable numerical solution for the div-curl system (1.1)-(1.3) with low H α -regularity (α > 0) assumption for the true solution u.…”

A new primal-dual weak Galerkin finite element method for ill-posed elliptic Cauchy problems

“…The PDWG framework provides mechanisms to enhance the stability of a numerical scheme by combining solutions of the primal and the dual (adjoint) equation. PDWG methods have been successfully applied to solve the second order elliptic equation in nondivergence form [47], the elliptic Cauchy problem [46,48], the Fokker-Planck type equation [49], the convection diffusion equation [13,54], and the transport equation [50,36]. The PDWG method has the following advantages over other existing schemes:…”

New Primal-Dual Weak Galerkin Finite Element Methods for Convection-Diffusion Problems

“…The convection-diffusion equations arise in many areas of science and engineering. Readers are referred to the "Introduction" Section in [21] and the references cited therein for a detailed description of the convection-diffusion equations.…”