Symmetric interior penalty Galerkin approaches for two-dimensional parabolic interface problems with low regularity solutions

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:330

Article

Song, Lunji¹  Zhao, Shan² 

[1] Lanzhou Univ, Sch Math & Stat, Gansu Key Lab Appl Math & Complex Syst, Lanzhou 730000, Gansu, Peoples R China

[2] Univ Alabama, Dept Math, Tuscaloosa, AL 35487 USA

关键词: Parabolic interface problems;    Interior penalty discontinuous Galerkin methods;    Low regularity solution;    Error estimates;    Stability;   

DOI  :  10.1016/j.cam.2017.09.018

来源: Elsevier

【 摘 要 】

This work presents novel finite element approaches for solving a parabolic partial differential equation with discontinuous coefficients and low regularity solutions in a bounded convex polyhedral domain. A spatial semi-discretization based on symmetric interior penalty Galerkin (SIPG) approximations is constructed and analyzed by using discontinuous piecewise linear functions. For smooth initial data, spatial errors in the broken L-2, H-1 and L-2(H-1) norms are proven to be optimal with respect to low regularity solutions, which are only piecewise H1+s smooth with 0 < s <= 1. Furthermore we present the full SIPG discretizations based on an Euler backward finite difference time discretization. The proposed approximations are shown to be unconditionally stable, and have nearly the optimal L-2(L-2) and L-2(H-1) error estimates, even when the regularities of the solutions are low on the whole domain. The error estimates are optimal with respect to time, sharply depending on the indexes in the global and local regularity. Numerical experiments for two-dimensional parabolic interface problems verify the theoretical convergence rates. (C) 2017 Elsevier B.V. All rights reserved.

【 授权许可】

Free