A fast iterative method for discretized Volterra–Fredholm integral equations

Solving the nonlinear systems arising in the discretization in space and time of Volterra–Fredholm integral equations by Newton iteration leads to dense linear systems whose dimension depends on the spatial mesh. The solution of these linear systems can hence be very costly. Here we try to reduce these costs by solving each Newton iteration by a non-stationary inner iteration process. Each inner iteration again requires the solution of a linear system. However, since the splitting matrix is diagonal, now the components or sets of components can be computed in parallel. The performance of this iteration method is illustrated by means of a few significative examples.