Convergence and asymptotic stability of Galerkin methods for linear parabolic equations with delays

This paper is concerned with the convergence and asymptotic stability of semidiscrete and full discrete schemes for linear parabolic equations with delay. These full discrete numerical processes include forward Euler, backward Euler and Crank-Nicolson schemes. The optimal convergence orders are consistent with those of the original parabolic equation. It is proved that the semidiscrete scheme, backward Euler and Crank-Nicolson full discrete schemes can unconditionally preserve the delay-independent asymptotic stability, but some additional restrictions on time and spatial stepsizes of the forward Euler full discrete scheme is needed to preserve the delay-independent asymptotic stability. Numerical experiments illustrate the theoretical results.