AN hp-VERSION DISCONTINUOUS GALERKIN METHOD FOR INTEGRO-DIFFERENTIAL EQUATIONS OF PARABOLIC TYPE

• Published in: SIAM journal on numerical analysis Vol. 49; no. 3/4; pp. 1369 - 1396
• Main Authors: MUSTAPHA, K., BRUNNER, H., MUSTAPHA, H., SCHÖTZAU, D.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
• Subjects: Acceleration of convergence; Algebra; Applied mathematics; Approximation; Degrees of polynomials; Error analysis; Error bounds; Error rates; Exact sciences and technology; Finite element method; Galerkin methods; Mathematical analysis; Mathematical discontinuity; Mathematics; Methods; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations; Perceptron convergence procedure; Polynomials; Sciences and techniques of general use; 65M15; Exponential convergence; Error analysis; Error estimation; weakly singular kernel; Optimal estimation; Finite element method; Parabolic equation; Numerical solution; 65M70; hp-version DG time-stepping; Convergence rate; 45J05; 65M50; 65M99; Integrodifferential equation; 45D05; Volterra equation; Volterra integral equations; Discontinuous Galerkin method; Convergence acceleration; Optimal convergence; parabolic Volterra integro-differential equation; Discrete scheme; Discretization method; Numerical analysis; Optimal approximation; 65M60; fully discrete scheme; Singular kernel
• ISSN: 0036-1429; 1095-7170
• DOI: 10.1137/100797114

We study the numerical solution of a class of parabolic 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.