Numerical analysis of a nonlocal Volterra integro-differential equation

Abstract

We study the numerical approximation to a nonlocal Volterra integro-differential equation, in which the integral term is the convolution product of a positive-definite kernel and a nonlocal peridynamic differential operator. Compared with the classical differential operators, the nonlocal peridynamic differential operators describe, e.g., discontinuities, and have domonstrated more widespread applications. The equation is discretized in space by the Galerkin finite element method, and we accordingly prove its error estimate. We then discretize the equation in time by the backward Euler method, and a positive quadrature rule is combined to approximate the convolution term. The convergence rate of the fully-discrete finite element scheme is proved, and numerical experiments are carried out to substantiate the theoretical findings.