Uniform stability of linear multistep methods in Galerkin procedures for parabolic problems

Abstract

Linear multistep methods are considered which have a stability region S and are D-stable on the whole boundary ∂S⊂S of S. Error estimates are derived which hold uniformly for the class of initial value problems Y′=AY+B(t), t>0, Y(0)=Y0 with normal matrix A satisfying the spectral condition Sp(ΔtA)⊂S, Δt time step, Sp(A) spectrum of A. Because of this property, the result can be applied to semidiscrete systems arising in the Galerkin approximation of parabolic problems. Using known results of the Ritz theory in elliptic boundary value problems error bounds for Galerkin multistep procedures are then obtained in this way.

Keywords

• numerical solution of ordinary differential equations
• A-priori error estimates of linear multistep methods
• Linear multistep methods with region of absolute stability
• uniform boundedness of powers of Frobenius matrices.