On a Full Discretisation for Nonlinear Second-Order Evolution Equations with Monotone Damping: Construction, Convergence, and Error Estimates

Abstract

Convergence of a full discretisation method is studied for a class of nonlinear second order in time evolution equations, where the nonlinear operator acting on the first-order time derivative of the solution is supposed to be hemicontinuous, monotone, coercive and to satisfy a certain growth condition, and the operator acting on the solution is assumed to be linear, bounded, symmetric, and strongly positive. The numerical approximation combines a Galerkin spatial discretisation with a novel time discretisation obtained from a reformulation of the second-order evolution equation as a first-order system and an application of the two-step backward differentiation formula with constant time stepsizes. Convergence towards the weak solution is shown for suitably chosen piecewise polynomial in time prolongations of the resulting fully discrete solutions, and an a priori error estimate ensures convergence of second order in time provided that the exact solution to the problem fulfils certain regularity requirements. A numerical example for a model problem describing the displacement of a vibrating membrane in a viscous medium illustrates the favourable error behaviour of the proposed full discretisation method in situations where regular solutions exist.