An Interior a Priori Estimate for Parabolic Difference Operators and an Application

Magnus Bondesson
1971 Mathematics of Computation  
A general class of finite-difference approximations to a parabolic system of differential equations in a bounded domain fi is considered. It is shown that if a solution Uh of the discrete problem converges in a discrete L1 norm to a solution U of the continuous problem as the mesh size A tends to zero, then the difference quotients of Uh converge to the corresponding derivatives of U, the convergence being uniform on any compact subset of Q. In particular, Uh converges uniformly on compact
more » ... mly on compact subsets to U as h tends to zero, provided there is convergence in the discrete L1 norm. The main part of the paper is devoted to the establishment of an a priori estimate for the solutions of the discrete problem. This estimate is then used to derive the stated result. Received May 15, 1970. AMS 1970 subject classifications. Primary 65N10, 65N15. Key words and phrases. Parabolic difference operator, a priori estimate, boundary value problem, convergence in maximum norm, convergence of difference quotients.
doi:10.2307/2005131 fatcat:ujns4abq7vcuzffojfam36vp2m