FINITE DIFFERENCE APPROXIMATION OF A PARABOLIC PROBLEM WITH VARIABLE COEFFICIENTS

Bosko S. Jovanovic, Zorica Milovanovic

Abstract: We study the convergence of a finite difference scheme that approximates the third initial-boundary-value problem for parabolic equation with variable coefficients on a unit square. We assume that the generalized solution of the problem belongs to the Sobolev space $W_2^{s,s/2}$, $ s\leq 3$. An almost second-order convergence rate estimate (with additional logarithmic factor) in the discrete $W^{1,1/2}_2$ norm is obtained. The result is based on some nonstandard a priori estimates involving fractional order discrete Sobolev norms.

Keywords: parabolic initial-boundary-value problem, oblique derivative boundary condition, finite differences, Sobolev spaces, convergence rate estimates

Classification (MSC2000): 65M15

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