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

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- by Kenneth Eriksson PDF
- Math. Comp.
**45**(1985), 109-142 Request permission

## Abstract:

We seek approximations of the solution*u*of the Neumann problem for the equation $Lu = f$ in $\Omega$ with special emphasis on high-order accuracy at a given point ${x_0} \in \bar \Omega$. Here $\Omega$ is a bounded domain in ${R^N}(N \geqslant 2)$ with smooth boundary, and

*L*is a second-order, uniformly elliptic, differential operator with smooth coefficients. An approximate solution ${u_h}$ is determined by the standard Galerkin method in a space of continuous piecewise polynomials of degree at most $r - 1$ on a partition ${\Delta _h}({x_0},\alpha )$ of $\Omega$. Here

*h*is a global mesh-size parameter, and $\alpha$ is the degree of a certain systematic refinement of the mesh around the given point ${x_0}$, where larger $\alpha$’s mean finer mesh, and $\alpha = 0$ corresponds to the quasi-uniform case with no refinement. It is proved that, for suitable (sufficiently large) $\alpha$’s the high-order error estimate $(u - {u_h})({x_0}) = O({h^{2r - 2}})$ holds. A corresponding estimate with the same order of convergence is obtained for the first-order derivatives of $u - {u_h}$. 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 ${x_0}$. 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 ${x_0}$ and square integrable in the rest of $\Omega$. The proof of this uses high-order negative-norm estimates of $u - {u_h}$. 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.

## References

- Robert A. Adams,
*Sobolev spaces*, Pure and Applied Mathematics, Vol. 65, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR**0450957** - 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 J. E. Osborn,
*Rate of convergence estimates for nonselfadjoint eigenvalue approximations*, Math. Comp.**27**(1973), 525–549. MR**366029**, DOI 10.1090/S0025-5718-1973-0366029-9 - J. H. Bramble and A. H. Schatz,
*Higher order local accuracy by averaging in the finite element method*, Math. Comp.**31**(1977), no. 137, 94–111. MR**431744**, DOI 10.1090/S0025-5718-1977-0431744-9 - Kenneth Eriksson,
*Improved accuracy by adapted mesh-refinements in the finite element method*, Math. Comp.**44**(1985), no. 170, 321–343. MR**777267**, DOI 10.1090/S0025-5718-1985-0777267-3 - Kenneth Eriksson,
*Improved accuracy by adapted mesh-refinements in the finite element method*, Math. Comp.**44**(1985), no. 170, 321–343. MR**777267**, DOI 10.1090/S0025-5718-1985-0777267-3 - 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** - 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
Ju. P. Krasovskiĭ, "Isolation of singularities of the Green’s function," - Joachim A. Nitsche,
*$L_{\infty }$-error analysis for finite elements*, Mathematics of finite elements and applications, III (Proc. Third MAFELAP Conf., Brunel Univ., Uxbridge, 1978) Academic Press, London-New York, 1979, pp. 173–186. MR**559297** - 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 - A. H. Schatz and L. B. Wahlbin,
*Maximum norm estimates in the finite element method on plane polygonal domains. I*, Math. Comp.**32**(1978), no. 141, 73–109. MR**502065**, DOI 10.1090/S0025-5718-1978-0502065-1 - A. H. Schatz and L. B. Wahlbin,
*Maximum norm estimates in the finite element method on plane polygonal domains. I*, Math. Comp.**32**(1978), no. 141, 73–109. MR**502065**, DOI 10.1090/S0025-5718-1978-0502065-1 - A. H. Schatz and L. B. Wahlbin,
*On the quasi-optimality in $L_{\infty }$ of the $\dot H^{1}$-projection into finite element spaces*, Math. Comp.**38**(1982), no. 157, 1–22. MR**637283**, DOI 10.1090/S0025-5718-1982-0637283-6 - Martin Schechter,
*On $L^{p}$ estimates and regularity. I*, Amer. J. Math.**85**(1963), 1–13. MR**188615**, DOI 10.2307/2373179 - Robert Schreiber,
*Finite element methods of high-order accuracy for singular two-point boundary value problems with nonsmooth solutions*, SIAM J. Numer. Anal.**17**(1980), no. 4, 547–566. MR**584730**, DOI 10.1137/0717047 - 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 - Vidar Thomée,
*High order local approximations to derivatives in the finite element method*, Math. Comp.**31**(1977), no. 139, 652–660. MR**438664**, DOI 10.1090/S0025-5718-1977-0438664-4

*Math. USSR-Izv.*, v. 1, 1967, pp. 935-966. A. Louis & F. Natterer,

*Acceleration of Convergence for Finite Element Solutions of the Poisson Equation on Irregular Meshes*, Preprint, 1977.

## Additional Information

- © Copyright 1985 American Mathematical Society
- Journal: Math. Comp.
**45**(1985), 109-142 - MSC: Primary 65N30
- DOI: https://doi.org/10.1090/S0025-5718-1985-0790647-5
- MathSciNet review: 790647