Time discretization of an integro-differential equation of parabolic type
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Discretization with variable time steps of an evolution equation with a positive-type memory term
Journal of Computational and Applied Mathematics
Numerical solution of parabolic integro-differential equations by the discontinuous Galerkin method
Mathematics of Computation
SIAM Journal on Numerical Analysis
hp-Discontinuous Galerkin Time-Stepping for Volterra Integrodifferential Equations
SIAM Journal on Numerical Analysis
A second-order accurate numerical method for a fractional wave equation
Numerische Mathematik
SIAM Journal on Numerical Analysis
Journal of Computational and Applied Mathematics
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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.