Finite element method for elliptic problems with edge singularities. (English) Zbl 0936.65130
The authors consider numerical approximations of tangentially regular solutions of the Dirichlet problem for homogeneous elliptic partial differential equations in three dimensions. The solution domain is a bounded polygonal domain in two dimensions extended infinitely in the third. Theoretical results employ Fourier transforms in the infinite direction with numerical results attained by truncated Fourier series. The bounded polygonal cross-section is treated using finite elements.
For this configuration, the solution can exhibit singularities arising from corners in the cross-section and yet maintain regularity in the third direction. This situation can lead to slow convergence rates. Convergence rates can be improved by augmenting the finite element spaces with certain singular functions. These singular functions are easily constructed in two dimensions but their extension to three dimensions is more delicate. The authors construct the singular functions first in two space and one Fourier dimension and then use inverse transforms to return to three space dimensions. The transform variable adds an infinite dimension to the normally finite-dimensional finite element spaces, but such important features as completeness remain true. Error estimates using the finite element method augmented with singular functions are proved and demonstrate the effectiveness of the method.
For this configuration, the solution can exhibit singularities arising from corners in the cross-section and yet maintain regularity in the third direction. This situation can lead to slow convergence rates. Convergence rates can be improved by augmenting the finite element spaces with certain singular functions. These singular functions are easily constructed in two dimensions but their extension to three dimensions is more delicate. The authors construct the singular functions first in two space and one Fourier dimension and then use inverse transforms to return to three space dimensions. The transform variable adds an infinite dimension to the normally finite-dimensional finite element spaces, but such important features as completeness remain true. Error estimates using the finite element method augmented with singular functions are proved and demonstrate the effectiveness of the method.
Reviewer: Myron Sussman (Bethel Park)
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 35J40 | Boundary value problems for higher-order elliptic equations |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
Keywords:
Fourier-finite element method; singular function method; regularizing kernel; error estimate; Dirichlet problem; Fourier transform; numerical results; finite elements; convergenceReferences:
| [1] | R.A. Adams, Sobolev Spaces, Academic Press, London, 1975.; R.A. Adams, Sobolev Spaces, Academic Press, London, 1975. · Zbl 0314.46030 |
| [2] | Apel, Th.; Dobrowolski, M., Anisotropic interpolation with applications to the finite element method, Computing, 47, 277-293 (1992) · Zbl 0746.65077 |
| [3] | Apel, Th.; Heinrich, B., Mesh refinement and windowing near edges for some elliptic problem, SIAM J. Numer. Anal., 31, 695-708 (1994) · Zbl 0807.65122 |
| [4] | Th. Apel, S. Nicaise, Elliptic problems in domains with edges: anisotropic regularity and anisotropic finite element meshes, in: J. Cea, D. Chenais, G. Geymonat, J.L. Lions (Eds.), Partial Differential Equations and Functional Analysis (In Memory of Pierre Grisvard), Partial Nonlinear Differential Equations, vol. 22, Birkhäuser, Boston, 1996, pp. 18-34.; Th. Apel, S. Nicaise, Elliptic problems in domains with edges: anisotropic regularity and anisotropic finite element meshes, in: J. Cea, D. Chenais, G. Geymonat, J.L. Lions (Eds.), Partial Differential Equations and Functional Analysis (In Memory of Pierre Grisvard), Partial Nonlinear Differential Equations, vol. 22, Birkhäuser, Boston, 1996, pp. 18-34. · Zbl 0854.35005 |
| [5] | Apel, Th.; Sändig, A.-M.; Whiteman, J. R., Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains, Math. Methods Appl. Sci., 19, 63-85 (1996) · Zbl 0838.65109 |
| [6] | Babuska, I., Finite element method for domains with corners, Computing, 6, 264-273 (1970) · Zbl 0224.65031 |
| [7] | I. Babuska, A.K. Aziz, 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.; I. Babuska, A.K. Aziz, 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. · Zbl 0268.65052 |
| [8] | H. Blum, Treatment of corners singularities in: E. Stein, W.L. Wendland (Eds.), Finite 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.; H. Blum, Treatment of corners singularities in: E. Stein, W.L. Wendland (Eds.), Finite 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. · Zbl 0658.00016 |
| [9] | Blum, H.; Dobrowolski, M., On finite element methods for elliptic equations on domains with corners, Computing, 28, 1, 53-63 (1982) · Zbl 0465.65059 |
| [10] | Bourlard, M.; Dauge, M.; Nicaise, S., Error estimates on the coefficients obtained by the singular function method, Numer. Funct. Anal. Optim., 10, 1077-1113 (1989) · Zbl 0705.65079 |
| [11] | M. Bourlard, M. Dauge, M.-S. Lubuma, S. Nicaise, Coefficients des singularités pour des problèmes aux limites elliptiques sur un domaine à points coniques I; II, RAIRO Modél. Math. Anal. Numér. 24 (1990) 27-52, 343-367.; M. Bourlard, M. Dauge, M.-S. Lubuma, S. Nicaise, Coefficients des singularités pour des problèmes aux limites elliptiques sur un domaine à points coniques I; II, RAIRO Modél. Math. Anal. Numér. 24 (1990) 27-52, 343-367. · Zbl 0723.35035 |
| [12] | Bourlard, M.; Dauge, M.; Lubuma, M.-S.; Nicaise, 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 |
| [13] | C. Canuto, Y. Hussaini, A. Quarteroni, T. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin, 1988.; C. Canuto, Y. Hussaini, A. Quarteroni, T. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin, 1988. · Zbl 0658.76001 |
| [14] | P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978.; P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. · Zbl 0383.65058 |
| [15] | Ph. Clément, Approximation by finite element functions using local regularization, RAIRO Anal. Numér. R2 (1975) 77-84.; Ph. Clément, Approximation by finite element functions using local regularization, RAIRO Anal. Numér. R2 (1975) 77-84. · Zbl 0368.65008 |
| [16] | M. Dauge, Elliptic Boundary Value Problems on Corner Domains, Lectures Notes in Mathematics, vol. 1341, Springer, Berlin, 1988.; M. Dauge, Elliptic Boundary Value Problems on Corner Domains, Lectures Notes in Mathematics, vol. 1341, Springer, Berlin, 1988. · Zbl 0668.35001 |
| [17] | Destuynder, P.; Djaoua, M., Estimation de l’erreur sur le coefficient de la singularité de la solution d’un problème elliptique sur un ouvert avec coin, RAIRO Anal. Numér., 14, 239-248 (1980) · Zbl 0456.65062 |
| [18] | M. Dobrowolski, Numerical Approximation of Elliptic Interface and Corner Problems, Habilitationsscrhift, Bonn, 1981.; M. Dobrowolski, Numerical Approximation of Elliptic Interface and Corner Problems, Habilitationsscrhift, Bonn, 1981. |
| [19] | Fix, G. J.; Gulati, S.; Wakoff, G. I., On the use of singular functions with finite element approximations, J. Comput. Phys., 13, 209-228 (1973) · Zbl 0273.35004 |
| [20] | V. Girault, P.A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer, Berlin, 1986.; V. Girault, P.A. Raviart, Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms, Springer, Berlin, 1986. · Zbl 0585.65077 |
| [21] | P. Grisvard, Singularités des problèmes aux limites dans des polyèdres, Séminaire Goulaouic Meyer Schwartz 8 (1981-82).; P. Grisvard, Singularités des problèmes aux limites dans des polyèdres, Séminaire Goulaouic Meyer Schwartz 8 (1981-82). · Zbl 0499.35038 |
| [22] | P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985.; P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985. · Zbl 0695.35060 |
| [23] | Grisvard, P., Edge behaviour of the solution of an elliptic problem, Math. Nachr., 132, 281-299 (1987) · Zbl 0639.35008 |
| [24] | Heinrich, B., The Fourier-finite-element-method for Poisson’s equation in axisymmetric domains with edges, SIAM J. Numer. Anal., 33, 1885-1911 (1996) · Zbl 0858.65108 |
| [25] | Kondratiev, V. A., Boundary value problems for elliptic equations in domains with conical or singular points, Trans. Moscow Math. Soc., 16, 227-313 (1967) · Zbl 0194.13405 |
| [26] | Kozlov, V. A.; Maz’ya, V. G., Spectral properties of the operator bundles generated by elliptic boundary value problems in a cone, Funct. Anal. Appl., 22, 114-121 (1988) · Zbl 0672.35050 |
| [27] | Lubuma, J.-M.-S.; Nicaise, S., Méthodes d’éléments finis raffinés pour le problème de Dirichlet dans un polyèdre, C.R. Acad. Sci. Paris, Série I, 315, 1207-1210 (1992) · Zbl 0782.65134 |
| [28] | Lubuma, J.-M.-S.; Nicaise, S., Dirichlet problems in polyhedral domains I: regularity of the solutions, Math. Nachr., 168, 243-261 (1994) · Zbl 0844.35014 |
| [29] | Lubuma, J.-M.-S.; Nicaise, S., Dirichlet problems in polyhedral domains II: approximations by FEM and BEM, J. Comput. Appl. Math., 61, 13-27 (1995) · Zbl 0840.65110 |
| [30] | J.-M.-S. Lubuma, S. Nicaise, Regularity of the solutions of Dirichlet problems in polyhedral domains in: M. Costabel, M. Dauge, S. Nicaise (Eds.), Boundary Value Problems and Integral Equations in Nonsmooth Domains, Lecture Notes in Pure and Applied Math., vol. 167, Marcel Dekker, New York, 1994, pp. 171-184.; J.-M.-S. Lubuma, S. Nicaise, Regularity of the solutions of Dirichlet problems in polyhedral domains in: M. Costabel, M. Dauge, S. Nicaise (Eds.), Boundary Value Problems and Integral Equations in Nonsmooth Domains, Lecture Notes in Pure and Applied Math., vol. 167, Marcel Dekker, New York, 1994, pp. 171-184. · Zbl 0822.35020 |
| [31] | Lubuma, J.-M.-S.; Nicaise, S., Méthode de fonctions singulières pour problèmes aux limites avec singularités d’ arêtes, C.R. Acad. Sci. Paris, Série I, 319, 1109-1114 (1994) · Zbl 0810.65090 |
| [32] | A. Maghnouji, Problèmes elliptiques et paraboliques dans des domaines non-réguliers Thesis, Univ. of Lille I, France, 1992.; A. Maghnouji, Problèmes elliptiques et paraboliques dans des domaines non-réguliers Thesis, Univ. of Lille I, France, 1992. |
| [33] | Maghnouji, A.; Nicaise, S., Coefficients of the singularities of elliptic and parabolic problems in domains with edges, Numer. Funct. Anal. Optim., 18, 805-825 (1997) · Zbl 0897.35025 |
| [34] | Maz’ya, V. G.; Plamenevskii, B. A., On the coefficients in the asymptotics of the solutions of an elliptic boundary value problem in domains with conical points, J. Soviet. Math., 9, 750-764 (1978) · Zbl 0396.35038 |
| [35] | Maz’ya, V. G.; Plamenevskii, B. A., Coefficients in the asymptotics of the solutions of an elliptic boundary value problem in a cone, Amer. Math. Soc. Transl., 123, 57-88 (1984) · Zbl 0554.35036 |
| [36] | Maz’ya, V. G.; Rossman, J., Uber die Asymptotik der Lösungen elliptischer Randwertaufgaben in der Umgebung von Kanten, Math. Nachr., 138, 27-53 (1988) · Zbl 0672.35020 |
| [37] | Mercier, B.; Raugel, G., Résolution d’un problème aux limites dans un ouvert axisymétrique par éléments finis en \(r,z\) et séries de Fourier en \(θ\), RAIRO Anal. Numér., 16, 405-461 (1982) · Zbl 0531.65054 |
| [38] | M. Moussaoui, Sur l’approximation des solutions du problème de Dirichlet dans un ouvert avec coins in: P. Grisvard, W. Wendland, J.R. Whiteman (Eds.), Singularities and constructive methods for their treatment, Lecture Notes in Mathematics, vol. 1121, Springer, Berlin, 1985, pp. 199-206.; M. Moussaoui, Sur l’approximation des solutions du problème de Dirichlet dans un ouvert avec coins in: P. Grisvard, W. Wendland, J.R. Whiteman (Eds.), Singularities and constructive methods for their treatment, Lecture Notes in Mathematics, vol. 1121, Springer, Berlin, 1985, pp. 199-206. · Zbl 0575.65102 |
| [39] | Stephan, E.; Whiteman, J. R., Singularities of the Laplacian at corners and edges of three-dimensional domains and their treatment with finite element methods, Math. Methods Appl. Sci., 10, 339-350 (1988) · Zbl 0671.35018 |
| [40] | Stephan, E.; von Petersdorff, T., Decompositions in edge and corner singularities for the solution of the Dirichlet problem of the Laplacian in a polyhedron, Math. Nachr., 149, 71-104 (1990) · Zbl 0780.35026 |
| [41] | G. Strang, J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973.; G. Strang, J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973. · Zbl 0356.65096 |
| [42] | R. Vichnevetsky, J.B. Bowles, Fourier Analysis of Numerical Approximations of Hyperbolic Equations, SIAM, Philidelphia, 1982.; R. Vichnevetsky, J.B. Bowles, Fourier Analysis of Numerical Approximations of Hyperbolic Equations, SIAM, Philidelphia, 1982. · Zbl 0495.65041 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
