Improving the rate of convergence of ‘high order finite elements’ on polygons and domains with cusps. (English) Zbl 1116.65119
The authors develop a new method for the construction of graded finite element meshes on a polygonal domain. This method leads to quasi-optimal rates of convergence for the finite element approximations of the Poisson problem \(-\Delta u=f\) with Dirichlet boundary conditions.
The main result can be summarized as follows: if we denote by \(u_{V}\in V\) the finite element approximation of \(u\), then \(\| u-u_{V}\| _{H^{1}}\leq C\text{dim}(V)^{-m/2}(\| f\| _{H^{m-1}})\). The constant \(C\) in this estimate is independent of \(V\) and \(\text{dim}(V)\), as \(\text{dim}(V)\rightarrow \infty \). Here \(m\geq 1\) is an integer, which could be thought of as the degree of piece-wise polynomials used in the finite element approximation.
The existence of such a sequence of spaces, was first proved by I. Babuska [Computing 6, 264–273 (1970; Zbl 0224.65031)]. The graded meshes considered in this paper generalize the ones constructed by G. Raugel [C. R. Acad. Sci., Paris, Sér. A 286, 791–794 (1978; Zbl 0377.65058)] and by I. Babuska, R. B. Kellogg, and J. Pitkäranta [Numer. Math. 33, 447–471 (1979; Zbl 0423.65057)]. The method of proof is based on a well posedness result for the Poisson equation in weighted Sobolev spaces and on the dilation properties of the weighted Sobolev spaces. The weight considered is the distance to the vertices. Numerical tests are provided.
The main result can be summarized as follows: if we denote by \(u_{V}\in V\) the finite element approximation of \(u\), then \(\| u-u_{V}\| _{H^{1}}\leq C\text{dim}(V)^{-m/2}(\| f\| _{H^{m-1}})\). The constant \(C\) in this estimate is independent of \(V\) and \(\text{dim}(V)\), as \(\text{dim}(V)\rightarrow \infty \). Here \(m\geq 1\) is an integer, which could be thought of as the degree of piece-wise polynomials used in the finite element approximation.
The existence of such a sequence of spaces, was first proved by I. Babuska [Computing 6, 264–273 (1970; Zbl 0224.65031)]. The graded meshes considered in this paper generalize the ones constructed by G. Raugel [C. R. Acad. Sci., Paris, Sér. A 286, 791–794 (1978; Zbl 0377.65058)] and by I. Babuska, R. B. Kellogg, and J. Pitkäranta [Numer. Math. 33, 447–471 (1979; Zbl 0423.65057)]. The method of proof is based on a well posedness result for the Poisson equation in weighted Sobolev spaces and on the dilation properties of the weighted Sobolev spaces. The weight considered is the distance to the vertices. Numerical tests are provided.
Reviewer: Nedyu Popivanov (Sofia)
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
References:
| [1] | Ammann, B., Lauter, R., Nistor, V.: Pseudodifferential operators on manifolds with a Lie structure at infinity. Math.DG/0304044 · Zbl 1133.58020 |
| [2] | Arnold, D.N., Boffi, D., Falk, R.S., Gastaldi, L.: Finite element approximation on quadrilateral meshes. Communications in Numerical Methods in Engineering 17, 805–812 (2001) · Zbl 0999.76073 · doi:10.1002/cnm.450 |
| [3] | Atiyah, M., Bott, R.: The index problem for manifolds with boundary. Differential Analysis (papers presented at the Bombay Colloquium 1964). Oxford University Press, 175–186 |
| [4] | Babuška, I., Aziz, A. K.:Survey lectures on the mathematical foundations of the finite element method. In: A.K. Aziz (Ed.), The mathematical foundations of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, London, 1972, pp 3–359 |
| [5] | Babuška, I., Aziz, A. K.:On the angle condition in the finite element method. SIAM J. Numer. Anal. 13 , 214–226 (1976) · Zbl 0324.65046 · doi:10.1137/0713021 |
| [6] | Babuška, I., Elman, H.C.: Performance of the h-p version of the finite element method with various elements. Pnt. J. Num. Methods Engrg. 36, 2503–1459 (1993) · Zbl 0780.73071 · doi:10.1002/nme.1620361502 |
| [7] | Babuška, I., Elman, H.C., Markle, K.: Parallel implementation of the h-p version of the finite element method on shared memory architecture. SIAM J. Numer. Anal. 13, 1433–1459 (1993) |
| [8] | Babuška, I.: Finite Element Method for Domains with Corners. Computing 6, 264–273 (1970) · Zbl 0224.65031 · doi:10.1007/BF02238811 |
| [9] | Babuška, I., Kellog, R.B., Pitkäranta, J.: Direct and inverse error estimates for finite elements with mesh refinements. Numer. Math. 33, 447–471 (1979) · Zbl 0423.65057 · doi:10.1007/BF01399326 |
| [10] | Babuška, I., Suri, M.: The P and H-P Versions of the finite element method, Basic principles and properties. SIAM Review 36, 578–632 (1994) · Zbl 0813.65118 · doi:10.1137/1036141 |
| [11] | Băcuţă, C., Bramble, J. H., Pasciak, J.: New interpolation results and applications to finite element methods for elliptic boundary value problems. Numerical Linear Algebra with Applications, Volume 10 (1–2), 33–64 (2003) |
| [12] | Băcuţă, C., Bramble, J. H., Xu, J.: Regularity estimates for elliptic boundary value problems in Besov spaces. Mathematics of Computation, 72, 1577–1595 (2003) · Zbl 1031.65133 |
| [13] | Băcuţă, C., Nistor, V., Zikatanov, L.: Regularity and well posedness for the Laplace operator on polyhedral domains. Math.AP/0410207 |
| [14] | Blum, H.: Treatment of corner singularities. In: E. Stein, W.L. Wendland, (eds.), Finie Element and Boundary Element Techniques from the Mathematical and Engineering Point of View, CISM Courses and Lectures, vol. 301, Springer, Wien, 1988, pp. 172–212 |
| [15] | Ciarlet, P.: Basic error estimates for elliptic problems. chapter in Handbook of Numerical Analysis (P. G. Ciarlet and J. L. Lions, eds.), Part 1, North Holland, (1991), 17–352. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, 1996 |
| [16] | Dauge, M.: Elliptic Boundary Value Problems on Corner Domains. Lecture Notes in Mathematics 1341, Springer-Verlag, Berlin, 1988 · Zbl 0668.35001 |
| [17] | Dauge, M., Nicaise, S., Bourland, M., Lubuma, M.S.: Coefficients des singularitiés pour des problèmes aux limities elliptiques sur un domaine à points coniques.I: Résultats généraux pour le problème de Dirichlet. RAIRO Modél. Math. Anal. Numér. 24, 27–52 (1990) |
| [18] | Dauge, M., Nicaise, S., Bourland, M., Lubuma, M.S.: Coefficients of the singularities for elliptic boundary value problems on domains with conical points III: Finite Element Methods on Polygonal Domains. SIAM J. Numer. Anal. 29, 136–155 (1992) · Zbl 0794.35015 · doi:10.1137/0729009 |
| [19] | DeVore, R., Lorentz, G.G.: Constructive Approximation. Grundlehren, Springer Verlag, New York, 1993 · Zbl 0797.41016 |
| [20] | DeVore, R., Dahlke, S.: Besov Regularity for elliptic boundary value problems. Comm. PDE 22 , 1–16 (1997) · Zbl 0883.35018 |
| [21] | Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston, 1985 · Zbl 0695.35060 |
| [22] | Grisvard, P.: Singularities in Boundary Value Problems. Masson, Paris, 1992 · Zbl 0766.35001 |
| [23] | Grubb, G.: Functional calculus of pseudodifferential boundary value problems. Second edition. Progress in Mathematics 65, Birkhäuser, Boston, 1996 · Zbl 0844.35002 |
| [24] | Kellogg, R. B.: Interpolation between subspaces of a Hilbert space. Technical note BN-719. Institute for Fluid Dynamics and Applied Mathematics, University of Maryland, College Park, 1971 · Zbl 0215.37504 |
| [25] | Kondratiev, V.A.: Boundary value problems for elliptic equations in domains with conical or angular points. Transl. Moscow Math. Soc. 16, 227–313 (1967) · Zbl 0194.13405 |
| [26] | Kozlov, V., Mazya, V., Rossmann, J.: Elliptic boundary value problems in domains with point singularities. Amer. Math. Soc., Providence, RI: 1997 · Zbl 0947.35004 |
| [27] | Kozlov, V., Mazya, V., Rossmann, J.: Spectral problems associated with corner singularities of solutions to elliptic equations. Amer. Math. Soc., Providence, RI: 2001 · Zbl 0965.35003 |
| [28] | Laubin, P.: High order convergence for collocation of second kind boundary integral equations on polygons. Numer. Math. 79, 107–140 (1998) · Zbl 0897.65073 · doi:10.1007/s002110050333 |
| [29] | Lewis, J., Parenti, C.: Pseudodifferential operators of Melin type. Comm. Part. Diff. Eq. 8, 477–544 (1983) · Zbl 0532.35085 · doi:10.1080/03605308308820276 |
| [30] | Lubuma, J., Nicaise, S.: Dirichlet problems in polyhedral domains II: Approximation by FEM and BEM. J. Comp. and Applied Math. 61, 13–27 (1995) · Zbl 0840.65110 · doi:10.1016/0377-0427(94)00050-B |
| [31] | Lubuma, J., Nicaise, S.: Finite element method for elliptic problems with edge singularities. J. Comp. and Applied Math. 106, 145–168 (1999) · Zbl 0936.65130 · doi:10.1016/S0377-0427(99)00062-X |
| [32] | Maz’ya, V., Roß mann, J.: Weighted Lp estimates of solutions to boundary value problems for second order elliptic systems in polyhedral domains. ZAMM Z. Angew. Math. Mech. 83, 435–467 (2003) · Zbl 1065.35109 · doi:10.1002/zamm.200310041 |
| [33] | Melrose, R.B.: The Atiyah-Patodi-Singer index theorem. Research Notes in Mathematics, 4, A.K. Peters, Ltd., MA 1993, xiv+377 · Zbl 0796.58050 |
| [34] | Melrose, R.B.: Geometric scattering theory. Stanford Lectures, Cambridge University Press, Cambridge, 1995 · Zbl 0849.58071 |
| [35] | Mitrea, M., Nistor, V.: Boundary layer potentials on manifolds with cylindrical ends. ESI preprint 2002 |
| [36] | Necas, J.: Les Methodes Directes en Theorie des Equations Elliptiques. Academia, Prague, 1967 |
| [37] | Nazarov, S.A., Plamenevsky, B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. Expositions in Mathematics 13, de Gruyter, New York, 1994 · Zbl 0806.35001 |
| [38] | Raugel, G.: Résolution numérique par une méthode d’éléments finis du problème de Dirichlet pour le laplacien dans un polygone. C.R. Acad. Sci. Paris 286 , 791–794 (1978) · Zbl 0377.65058 |
| [39] | Raugel, G.: Résolution numérique de problèmes elliptiques dans domaines avec coins. Thesis, University of Rennes, 1978 |
| [40] | Schatz, A. H., Wahlbin, L. B.: On the Quasi-Optimality in Lof the -Projection into Finite Element Spaces. Math. Comp. 38, 1–22 (1982) · Zbl 0483.65006 |
| [41] | Schatz, A. H., Sloan, I., Wahlbin, L. B.: Superconvergence in finite element methods and meshes which are locally symmetric with respect to a point. SIAM Journal of Numerical Analysis, to appear · Zbl 0855.65115 |
| [42] | Schrohe, E.: Fréchet algebra technique for boundary value problems on noncompact manifolds: Fredholm criteria and functional calculus via spectral invariance. Math. Nach. 1999, 145–185 · Zbl 0923.58054 |
| [43] | Taylor, M. E.: Partial differential equations, Applied Mathematical Sciences, vol. I, Springer-Verlag, New York, (1995–1997) |
| [44] | Taylor, M. E.: Partial differential equations, Applied Mathematical Sciences, vol. II, Springer-Verlag, New York, (1995–1997) |
| [45] | Verfurth, V.: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, 1996 (150 pp) · Zbl 1189.76394 |
| [46] | Wahlbin, L. B.: Local Behavior in Finite Element Methods. chapter in Handbook of Numerical Analysis (P. G. Ciarlet and J. L. Lions, eds.), Part 1, North Holland, (1991), 353–522. A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, 1996 (150 pp) |
| [47] | Wahlbin, L. B.: Superconvergence in Galerkin Finite Element Methods. Springer Lecture Notes in Mathematics 1605, Springer-Verlag New York, 1995 · Zbl 0826.65092 |
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