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
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.