Department of Mathematics
An hp-version discontinuous galerkin method for integro-differential equations of parabolic type
We study the numerical solution of a class of pa rabolic integro-differential equations with weakly singular kernels. We use an hp-version discontinuous Galerkin (DG) method for the discretization in time. We derive optimal hp-version error estimates and show that exponential rates of convergence can be achieved for solutions with singular (temporal) behavior near t = 0 caused by the weakly singular kernel. Moreover, we prove that by using nonuniformly refined time steps, optimal algebraic convergence rates can be achieved for the h-version DG method. We then combine the DG time-stepping method with a standard finite element discretization in space, and present an optimal error analysis of the resulting fully discrete scheme. Our theoretical results are numerically validated in a series of test problems. © 2011 Society for Industrial and Applied Mathematics.
Exponential convergence, Finite element method, Fully discrete scheme, Hp-version DG time-stepping, Parabolic volterra integro-differential equation, Weakly singular kernel
Source Publication Title
SIAM Journal on Numerical Analysis
Society for Industrial and Applied Mathematics
Link to Publisher's Edition
Mustapha, K., H. Brunner, H. Mustapha, and D. Schotzau. "An hp-version discontinuous galerkin method for integro-differential equations of parabolic type." SIAM Journal on Numerical Analysis 49.4 (2011): 1369-1396.