An interior a priori estimate for parabolic difference operators and an application

Author:
Magnus Bondesson

Journal:
Math. Comp. **25** (1971), 43-58

MSC:
Primary 65.68

MathSciNet review:
0290598

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A general class of finite-difference approximations to a parabolic system of differential equations in a bounded domain is considered. It is shown that if a solution of the discrete problem converges in a discrete norm to a solution *U* of the continuous problem as the mesh size *h* tends to zero, then the difference quotients of converge to the corresponding derivatives of *U*, the convergence being uniform on any compact subset of . In particular, converges uniformly on compact subsets to *U* as *h* tends to zero, provided there is convergence in the discrete 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.

**[1]**Avner Friedman,*Partial differential equations of parabolic type*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR**0181836****[2]**K. O. Friedrichs,*On the differentiability of the solutions of linear elliptic differential equations*, Comm. Pure Appl. Math.**6**(1953), 299–326. MR**0058828****[3]**S. Håkangård, "Some results on the rate of convergence for discrete initial-value problems," Report, Department of Mathematics, Chalmers Institute of Technology and the University of Goteborg.**[4]**Fritz John,*On integration of parabolic equations by difference methods. I. Linear and quasi-linear equations for the infinite interval*, Comm. Pure Appl. Math.**5**(1952), 155–211. MR**0047885****[5]**M. L. Juncosa and D. M. Young,*On the order of convergence of solutions of a difference equation to a solution of the diffusion equation*, J. Soc. Indust. Appl. Math.**1**(1953), 111–135. MR**0060907****[6]**M. L. Juncosa and David Young,*On the Crank-Nicolson procedure for solving parabolic partial differential equations*, Proc. Cambridge Philos. Soc.**53**(1957), 448–461. MR**0088804****[7]**Jörgen Löfström,*Besov spaces in theory of approximation*, Ann. Mat. Pura Appl. (4)**85**(1970), 93–184. MR**0267332****[8]**Jaak Peetre and Vidar Thomée,*On the rate of convergence for discrete initial-value problems*, Math. Scand.**21**(1967), 159–176 (1969). MR**0255085****[9]**S. L. Sobolev, "On estimates for certain sums for functions defined on a grid,"*Izv. Akad. Nauk SSSR Ser. Mat.*, v. 4, 1940, pp. 5-16. (Russian) MR**1**, 298.**[10]**Vidar Thomée,*Parabolic difference operators*, Math. Scand.**19**(1966), 77–107. MR**0209693****[11]**Vidar Thomée and Bertil Westergren,*Elliptic difference equations and interior regularity*, Numer. Math.**11**(1968), 196–210. MR**0224303****[12]**Olof B. Widlund,*On the rate of convergence for parabolic difference schemes. II*, Comm. Pure Appl. Math.**23**(1970), 79–96. MR**0264868**

Retrieve articles in *Mathematics of Computation*
with MSC:
65.68

Retrieve articles in all journals with MSC: 65.68

Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1971-0290598-9

Keywords:
Parabolic difference operator,
a priori estimate,
boundary value problem,
convergence in maximum norm,
convergence of difference quotients

Article copyright:
© Copyright 1971
American Mathematical Society