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

Eriksson, Kenneth

doi:10.1090/S0025-5718-1985-0777267-3

Improved accuracy by adapted mesh-refinements in the finite element method
HTML articles powered by AMS MathViewer

by Kenneth Eriksson PDF

Math. Comp. 44 (1985), 321-343 Request permission

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 $Lu = \delta$, where $\delta$ is the Dirac distribution.

References

Robert A. Adams, Sobolev spaces, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0450957
Ivo Babuška, Error-bounds for finite element method, Numer. Math. 16 (1970/71), 322–333. MR 288971, DOI 10.1007/BF02165003

The Mathematical Foundations of the Finite Element Method with Applications to Parital Differential Equations

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 263214, DOI 10.1137/0707006
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
Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
Philippe G. Ciarlet, Discrete variational Green’s function. I, Aequationes Math. 4 (1970), 74–82. MR 273838, DOI 10.1007/BF01817748

Improved Convergence by Mesh-Refinement in the Finite Element Method

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 331715, DOI 10.1090/S0025-5718-1973-0331715-3

Math. USSR-Izv.

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
Joachim A. Nitsche and Alfred H. Schatz, Interior estimates for Ritz-Galerkin methods, Math. Comp. 28 (1974), 937–958. MR 373325, DOI 10.1090/S0025-5718-1974-0373325-9
A. H. Schatz and L. B. Wahlbin, Interior maximum norm estimates for finite element methods, Math. Comp. 31 (1977), no. 138, 414–442. MR 431753, DOI 10.1090/S0025-5718-1977-0431753-X
Ridgway Scott, Finite element convergence for singular data, Numer. Math. 21 (1973/74), 317–327. MR 337032, DOI 10.1007/BF01436386
Ridgway Scott, Optimal $L^{\infty }$ estimates for the finite element method on irregular meshes, Math. Comp. 30 (1976), no. 136, 681–697. MR 436617, DOI 10.1090/S0025-5718-1976-0436617-2