Adaptive discretization of an integro-differential equation with a weakly singular convolution kernel Original Research Article

An integro-differential equation involving a convolution integral with a weakly singular kernel is considered. The kernel can be that of a fractional integral. The integro-differential equation is discretized using the discontinuous Galerkin method with piecewise constant basis functions. Sparse quadrature is introduced for the convolution term to overcome the problem with the growing amount of data that has to be stored and used in each time-step. A priori and a posteriori error estimates are proved. An adaptive strategy based on the a posteriori error estimate is developed. Finally, the precision and effectiveness of the algorithm are demonstrated in the case that the convolution is a fractional integral. This is done by comparing the numerical solutions with analytical solutions.