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

Abstract

We study decay properties of the numerical solutions of a class of partial differential equations

$$\begin{aligned} \ddot{u}(t)+Au(t)-\int \limits _{0}^{t}\beta (t-s)Au(s)ds=0,\quad t>0,\,u(0)=u_{0},\,\dot{u}(0)=u_{1}, \end{aligned}$$

which arises in the theory of linear viscoelasticity. Here $A$ is a positive self-adjoint densely defined linear operator in a Hilbert space $\mathbf {H}$, $u_{0},\,u_{1}\in \mathbf {H}$ and the real-valued kernel $\beta (t)$ is assumed to be nonnegative non-increasing, not identically $0$, and satisfy $\int _{0}^{\infty }\beta (t)dt< 1$. The proposed discretization uses convolution quadrature based on the trapezoidal rule in time, and piecewise linear finite elements in space. We establish the uniform $l_{t}^{\infty }(0,\infty ;\,\mathbf {H}) \bigcap$ $l_{t}^{1}(0,\infty ;\mathbf {H})$ stability numerical schemes, and Polynomial decay numerical methods in time. The fully discrete uniform $l_{t}^{\infty }(0,\infty ;\,\mathbf {H})$ error estimates are derived. Some simple numerical examples illustrate our theoretical error bounds.