Convergence characteristics of the Crank-Nicolson-Galerkin scheme for linear parabolic systems

Abstract

This paper is concerned with the investigation on the stability and convergence characteristics of the Crank-Nicolson-Galerkin scheme that is widely being employed for the numerical approximation of parabolic-type partial differential equations. Here, we present the theoretical analysis on its consistency and convergence, and we carry out the numerical experiments to examine the effect of the time-step sizeΔ t on theh- and ρ-convergence rates for various mesh sizesh and approximation ordersρ. We observed that the optimal convergence rates are achieved only when Δt,h andρ are chosen such that the total error is not affected by the oscillation behavior. In such case,Δ t is in linear relation with DOF, and furthermore its size depends on the singularity intensity of problems.