Decay Properties for the Numerical Solutions of a Partial Differential Equation with Memory

We study decay properties of the numerical solutions of a class of partial differential equations which arises in the theory of linear viscoelasticity. Here is a positive self-adjoint densely defined linear operator in a Hilbert space , and the real-valued kernel is assumed to be nonnegative non-increasing, not identically , and satisfy . The proposed discretization uses convolution quadrature based on the trapezoidal rule in time, and piecewise linear finite elements in space. We establish the uniform stability numerical schemes, and Polynomial decay numerical methods in time. The fully discrete uniform error estimates are derived. Some simple numerical examples illustrate our theoretical error bounds.