Improved accuracy by adapted mesh-refinements in the finite element method

Author:
Kenneth Eriksson

Journal:
Math. Comp. **44** (1985), 321-343

MSC:
Primary 65N30; Secondary 65N50

DOI:
https://doi.org/10.1090/S0025-5718-1985-0777267-3

MathSciNet review:
777267

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For appropriately adapted mesh-refinements, optimal order error estimates are proved for the finite element approximate solution of the Neumann problem for the second-order elliptic equation , where is the Dirac distribution.

**[1]**Robert A. Adams,*Sobolev spaces*, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR**0450957****[2]**Ivo Babuška,*Error-bounds for finite element method*, Numer. Math.**16**(1970/1971), 322–333. MR**0288971**, https://doi.org/10.1007/BF02165003**[3]**I. Babuška & A. K. Aziz, "Survey lectures on the mathematical foundations of the finite element method," Section 6.3.6 of*The Mathematical Foundations of the Finite Element Method with Applications to Parital Differential Equations*, (A. K. Aziz, ed.), Academic Press, New York, 1973.**[4]**J. H. Bramble and S. R. Hilbert,*Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation*, SIAM J. Numer. Anal.**7**(1970), 112–124. MR**0263214**, https://doi.org/10.1137/0707006**[5]**J. H. Bramble and A. H. Schatz,*Estimates for spline projections*, Rev. Française Automat. Informat. Recherche Opérationnelle Sér.**10**(1976), no. R-2, 5–37. MR**0436620****[6]**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****[7]**Philippe G. Ciarlet,*Discrete variational Green’s function. I*, Aequationes Math.**4**(1970), 74–82. MR**0273838**, https://doi.org/10.1007/BF01817748**[8]**K. Eriksson,*Improved Convergence by Mesh-Refinement in the Finite Element Method*, Thesis, Chalmers University of Technology and the University of Göteborg, 1981.**[9]**Stephen Hilbert,*A mollifier useful for approximations in Sobolev spaces and some applications to approximating solutions of differential equations*, Math. Comp.**27**(1973), 81–89. MR**0331715**, https://doi.org/10.1090/S0025-5718-1973-0331715-3**[10]**Ju. P. Krasovskii, "Isolation of singularities of the Green's function,"*Math. USSR-Izv.*, v. 1, 1967, pp. 935-966.**[11]**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****[12]**Joachim A. Nitsche and Alfred H. Schatz,*Interior estimates for Ritz-Galerkin methods*, Math. Comp.**28**(1974), 937–958. MR**0373325**, https://doi.org/10.1090/S0025-5718-1974-0373325-9**[13]**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**, https://doi.org/10.1090/S0025-5718-1977-0431753-X**[14]**Ridgway Scott,*Finite element convergence for singular data*, Numer. Math.**21**(1973/74), 317–327. MR**0337032**, https://doi.org/10.1007/BF01436386**[15]**Ridgway Scott,*Optimal 𝐿^{∞} estimates for the finite element method on irregular meshes*, Math. Comp.**30**(1976), no. 136, 681–697. MR**0436617**, https://doi.org/10.1090/S0025-5718-1976-0436617-2

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

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

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1985-0777267-3

Keywords:
Neumann problem,
Green's function,
finite element method,
mesh-refinement,
error estimate

Article copyright:
© Copyright 1985
American Mathematical Society