Finite element methods of optimal order for problems with singular data

Author:
Kenneth Eriksson

Journal:
Math. Comp. **44** (1985), 345-360

MSC:
Primary 65N30; Secondary 65N50

MathSciNet review:
777268

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An adapted finite element method is proposed for a class of elliptic problems with singular data. The idea is to subtract the main singularity from the solution and to solve for the remainder using suitable mesh-refinements. Optimal order error estimates are proved.

**[1]**Ivo Babuška,*Error-bounds for finite element method*, Numer. Math.**16**(1970/1971), 322–333. MR**0288971****[2]**Ivo Babuška and A. K. Aziz,*Survey lectures on the mathematical foundations of the finite element method*, The mathematical foundations of the finite element method with applications to partial differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York, 1972, pp. 1–359. With the collaboration of G. Fix and R. B. Kellogg. MR**0421106****[3]**J. H. Bramble and A. H. Schatz,*Estimates for spline projections*, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. \jname RAIRO Analyse Numérique**10**(1976), no. R-2, 5–37. MR**0436620****[4]**Philippe G. Ciarlet,*The finite element method for elliptic problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR**0520174****[5]**Philippe G. Ciarlet,*Discrete variational Green’s function. I*, Aequationes Math.**4**(1970), 74–82. MR**0273838****[6]**K. Eriksson,*Improved Convergence by Mesh-Refinement in the Finite Element Method*, Thesis, Chalmers University of Technology and the University of Göteborg, 1981.**[7]**Kenneth Eriksson,*Improved accuracy by adapted mesh-refinements in the finite element method*, Math. Comp.**44**(1985), no. 170, 321–343. MR**777267**, 10.1090/S0025-5718-1985-0777267-3**[8]**Ju. P. Krasovskii, "Isolation of singularities of the Green's function,"*Math. USSR-Izv.*, v. 1, 1967, pp. 935-966.**[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]**A. H. Schatz and L. B. Wahlbin,*Interior maximum norm estimates for finite element methods*, Math. Comp.**31**(1977), no. 138, 414–442. MR**0431753**, 10.1090/S0025-5718-1977-0431753-X**[11]**Ridgway Scott,*Finite element convergence for singular data*, Numer. Math.**21**(1973/74), 317–327. MR**0337032**

Retrieve articles in *Mathematics of Computation*
with MSC:
65N30,
65N50

Retrieve articles in all journals with MSC: 65N30, 65N50

Additional Information

DOI:
http://dx.doi.org/10.1090/S0025-5718-1985-0777268-5

Keywords:
Neumann problem,
Green's function,
adapted finite element methods,
mesh-refinement,
optimal order,
error estimate

Article copyright:
© Copyright 1985
American Mathematical Society