Estimates near plane portions of the boundary for discrete elliptic boundary problems

Author:
C. G. L. Johnson

Journal:
Math. Comp. **28** (1974), 909-935

MSC:
Primary 65N15

MathSciNet review:
0362943

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider an elliptic difference operator together with certain boundary difference operators near a plane portion of the boundary parallel to some coordinate direction. We prove discrete analogues of known estimates in and Schauder norms for elliptic boundary problems. The discrete estimates are then used to prove results about convergence near plane portions of the boundary of difference quotients of solutions of a discrete elliptic problem to the derivatives of the solution *u* of the corresponding continuous problem, when it is known that converges to *u* in the maximum norm or in a discrete norm as *h* tends to zero.

**[1]**S. Agmon, A. Douglis, and L. Nirenberg,*Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I*, Comm. Pure Appl. Math.**12**(1959), 623–727. MR**0125307****[2]**Leif Arkeryd,*On the 𝐿^{𝑝} estimates for elliptic boundary problems*, Math. Scand.**19**(1966), 59–76. MR**0224970****[3]**Magnus Bondesson,*Interior a priori estimates in discrete 𝐿_{𝑝} norms for solutions of parabolic and elliptic difference equations*, Ann. Mat. Pura Appl. (4)**95**(1973), 1–43. MR**0338566****[4]**J. H. Bramble and B. E. Hubbard,*A finite difference analogue of the Neumann problem for Poisson’s equation*, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal.**2**(1965), 1–14. MR**0191107****[5]**J. H. Bramble, B. E. Hubbard, and Vidar Thomée,*Convergence estimates for essentially positive type discrete Dirichlet problems*, Math. Comp.**23**(1969), 695–709. MR**0266444**, 10.1090/S0025-5718-1969-0266444-7**[6]**R. GRIGORIEFF,*Über die Koerzivität linearer elliptischer Differenzenoperatoren unter allgemeinen Randebedingungen*, Thesis, Frankfurt am Main, 1967.**[7]**Lars Hörmander,*On the regularity of the solutions of boundary problems*, Acta Math.**99**(1958), 225–264. MR**0131655****[8]**C. G. L. Johnson,*Estimates near plane portions of the boundary for discrete elliptic boundary problems*, Math. Comp.**28**(1974), 909–935. MR**0362943**, 10.1090/S0025-5718-1974-0362943-X**[9]**J.-L. Lions and E. Magenes,*Problèmes aux limites non homogènes et applications. Vol. 1*, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR**0247243****[10]**Jörgen Löfström,*Besov spaces in theory of approximation*, Ann. Mat. Pura Appl. (4)**85**(1970), 93–184. MR**0267332****[11]**J. PEETRE,*Reflexions on Besov Spaces*, Unpublished manuscript, Lund, 1966. (In Swedish).**[12]**David G. Schaeffer,*Approximation of the Dirichlet problem on a half space*, Acta Math.**129**(1972), no. 3–4, 281–295. MR**0395261****[13]**Vidar Thomée,*Discrete interior Schauder estimates for elliptic difference operators.*, SIAM J. Numer. Anal.**5**(1968), 626–645. MR**0238505****[14]**Vidar Thomée,*Convergence near plane boundaries of some elliptic difference schemes*, Numer. Math.**17**(1971), 45–53. MR**0295604****[15]**Vidar Thomée and Bertil Westergren,*Elliptic difference equations and interior regularity*, Numer. Math.**11**(1968), 196–210. MR**0224303****[16]**Vidar Thomée,*Elliptic difference operators and Dirichlet’s problem*, Contributions to Differential Equations**3**(1964), 301–324. MR**0163444****[17]**Miloš Zlámal,*Asymptotic error estimates in solving elliptic equations of the fourth order by the method of finite differences*, J. Soc. Indust. Appl. Math. Ser. B Numer. Anal.**2**(1965), 337–344. MR**0184446**

Retrieve articles in *Mathematics of Computation*
with MSC:
65N15

Retrieve articles in all journals with MSC: 65N15

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1974-0362943-X

Keywords:
Elliptic difference operator,
discrete elliptic boundary problem,
discrete and Schauder estimates,
convergence of difference quotients

Article copyright:
© Copyright 1974
American Mathematical Society