High-order local rate of convergence by mesh-refinement in the finite element method

Author:
Kenneth Eriksson

Journal:
Math. Comp. **45** (1985), 109-142

MSC:
Primary 65N30

DOI:
https://doi.org/10.1090/S0025-5718-1985-0790647-5

MathSciNet review:
790647

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Abstract: We seek approximations of the solution *u* of the Neumann problem for the equation in with special emphasis on high-order accuracy at a given point . Here is a bounded domain in with smooth boundary, and *L* is a second-order, uniformly elliptic, differential operator with smooth coefficients. An approximate solution is determined by the standard Galerkin method in a space of continuous piecewise polynomials of degree at most on a partition of . Here *h* is a global mesh-size parameter, and is the degree of a certain systematic refinement of the mesh around the given point , where larger 's mean finer mesh, and corresponds to the quasi-uniform case with no refinement. It is proved that, for suitable (sufficiently large) 's the high-order error estimate holds. A corresponding estimate with the same order of convergence is obtained for the first-order derivatives of . These estimates are sharp in the sense that the required degree of refinement in each case is essentially the same as is needed for the local approximation to this order near . For the estimates to hold, it is sufficient that the exact solution *u* have derivatives to the *r*th order which are bounded close to and square integrable in the rest of . The proof of this uses high-order negative-norm estimates of . The number of elements in the considered partitions is of the same order as in the corresponding quasi-uniform ones. Applications of the results to other types of boundary value problems are indicated.

**[1]**R. A. Adams,*Sobolev Spaces*. Academic Press, New York, 1975. MR**0450957 (56:9247)****[2]**J. H. Bramble & S. R. Hilbert, "Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation,"*SIAM J. Numer. Anal.*, 7, 1970, pp. 112-124. MR**0263214 (41:7819)****[3]**J. H. Bramble & J. E. Osborn, "Rate of convergence estimates for nonselfadjoint eigenvalue approximations,"*Math. Comp.*, v. 27, 1973, pp. 525-549. MR**0366029 (51:2280)****[4]**J. H. Bramble & A. H. Schatz, "Higher order local accuracy by averaging in the finite element method,"*Math. Comp.*, v. 31, 1977, pp. 94-111. MR**0431744 (55:4739)****[5]**K. Eriksson, "Improved accuracy by adapted mesh-refinements in the finite element method,"*Math. Comp.*, v. 44, 1985, pp. 321-343. MR**777267 (86j:65155a)****[6]**K. Eriksson, "Finite element methods of optimal order for problems with singular data,"*Math. Comp.*, v. 44, 1985, pp. 345-360. MR**777268 (86j:65155b)****[7]**P. G. Ciarlet,*The Finite Element Method for Elliptic Problems*, North-Holland, Amsterdam, 1978. MR**0520174 (58:25001)****[8]**S. R. Hilbert, "A mollifier useful for approximations in Sobolev spaces and some applications to approximating solutions of differential equations,"*Math. Comp.*, v. 27, 1973, pp. 81-89. MR**0331715 (48:10047)****[9]**Ju. P. Krasovskiĭ, "Isolation of singularities of the Green's function,"*Math. USSR-Izv.*, v. 1, 1967, pp. 935-966.**[10]**A. Louis & F. Natterer,*Acceleration of Convergence for Finite Element Solutions of the Poisson Equation on Irregular Meshes*, Preprint, 1977.**[11]**J. A. Nitsche, -*Error Analysis for Finite Elements*, Proc. 3rd Conf. on Mathematics of Finite Elements and Applications, Brunel University, 1978. MR**559297 (81f:65084)****[12]**J. A. Nitsche & A. H. Schatz, "Interior estimates for Ritz-Galerkin methods,"*Math. Comp.*, v. 28, 1974, pp. 937-958. MR**0373325 (51:9525)****[13]**A. H. Schatz & L. B. Wahlbin, "Interior maximum norm estimates for finite element methods,"*Math. Comp.*, v. 31, 1977, pp. 414-442. MR**0431753 (55:4748)****[14]**A. H. Schatz & L. B. Wahlbin, "Maximum norm estimates in the finite element method on plane polygonal domains. Part 1,"*Math. Comp.*, v. 32, 1978, pp. 73-109. MR**0502065 (58:19233a)****[15]**A. H. Schatz & L. B. Wahlbin, "Maximum norm estimates in the finite element method on plane polygonal domains. Part 2: Refinements,"*Math. Comp.*, v. 33, 1979, pp. 465-492. MR**0502067 (58:19233b)****[16]**A. H. Schatz & L. B. Wahlbin, "On the quasi-optimality in of the -projection into finite element spaces,"*Math. Comp.*, v. 38, 1982, pp. 1-22. MR**637283 (82m:65106)****[17]**M. Schechter, "On estimates and regularity. I."*Amer. J. Math.*, v. 85, 1963, pp. 1-13. MR**0188615 (32:6051)****[18]**R. Schreiber, "Finite element methods of high-order accuracy for singular two-point boundary value problems with nonsmooth solutions,"*SIAM J. Numer. Anal.*, v. 17, 1980, pp. 547-566. MR**584730 (82b:65139)****[19]**R. Scott, "Optimal estimates for the finite element method on irregular meshes,"*Math. Comp.*, v. 30, 1976, pp. 681-697. MR**0436617 (55:9560)****[20]**V. Thomée, "High order local approximations to derivatives in the finite element method,"*Math. Comp.*, v. 31, 1977, pp. 652-660. MR**0438664 (55:11572)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1985-0790647-5

Keywords:
Finite element method,
Galerkin approximation,
mesh-refinement,
high-order local convergence,
Neumann problem

Article copyright:
© Copyright 1985
American Mathematical Society