Mixed Weak Galerkin Method for Heat Equation with Random Initial Condition

Abstract

This paper is devoted to the numerical analysis of weak Galerkin mixed finite element method (WGMFEM) for solving a heat equation with random initial condition. To set up the finite element spaces, we choose piecewise continuous polynomial functions of degree j+1 with j≥0 for the primary variables and piecewise discontinuous vector-valued polynomial functions of degree j for the flux ones. We further establish the stability analysis of both semidiscrete and fully discrete WGMFE schemes. In addition, we prove the optimal order convergence estimates in L2 norm for scalar solutions and triple-bar norm for vector solutions and statistical variance-type convergence estimates. Ultimately, we provide a few numerical experiments to illustrate the efficiency of the proposed schemes and theoretical analysis.

A Stabilizer Free Weak Galerkin Finite Element Method for Elliptic Equation with Lower Regularity

“…The WG method is first introduced by Wang and Ye [17,18] for the second-order elliptic equations, and a stabilizer term is added to WG-FEM in order to enforce the connection of discontinuous functions across element boundaries [10,11]. Then the WG method finds applications in diverse areas including elliptic equations [9,27], parabolic equations [30,31,34,35], second-order linear wave equation [6], reaction-diffusion equations [8], Stokes equations [13,16,32], Maxwell equations [12,14,20], biharmonic equation [33], Cahn-Hilliard-Cook equation [5], stochastic parabolic equations [36,37], eigenvalue problems [28,29], and so on. Nevertheless, the stabilizer makes the finite element formulations and programming complex.…”

Weak Galerkin finite element method for linear poroelasticity problems

“…In analogy with discrete weak gradient, discrete weak divergence is introduced in [31] where the proposed WG mixed finite element method is applicable for general finite element partitions consisting of shape regular polygons in 2D or polyhedra in 3D. At present, WG method has been widely used to solve various partial differential equations, such as Helmholtz equations [19,9,33], linear parabolic equations [17,37,36], Biharmonic problems [18,7], Stokes problems [22], stochastic partial differential equations [40,41], and heat equations [39,38] etc.…”