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 c 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) Y with normal matrix A satisfying the spectral condition Sp(AtA) S At O 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. KEY WORDS AND PHRASES. Numerical solution of ordinary differential equation, A-pror error estimates of linear mtistep methods, Linear mbistep methods with region of absolute stability, Uniform boundedness of powers of Frobenius matrices