Previous |  Up |  Next

Article

Keywords:
elliptic boundary value problem; a priori error estimates; interpolation of non-smooth functions; finite element error; non-convex domains; edge singularities; anisotropic mesh grading
Summary:
An $L^2$-estimate of the finite element error is proved for a Dirichlet and a Neumann boundary value problem on a three-dimensional, prismatic and non-convex domain that is discretized by an anisotropic tetrahedral mesh. To this end, an approximation error estimate for an interpolation operator that is preserving the Dirichlet boundary conditions is given. The challenge for the Neumann problem is the proof of a local interpolation error estimate for functions from a weighted Sobolev space.
References:
[1] Apel, Th.: Interpolation of non-smooth functions on anisotropic finite element meshes. M2AN, Math. Model. Numer. Anal. (1999), 33 1149-1185. DOI 10.1051/m2an:1999139 | MR 1736894 | Zbl 0984.65113
[2] Apel, Th., Dobrowolski, M.: Anisotropic interpolation with applications to the finite element method. Computing 47 (1992), 277-293. DOI 10.1007/BF02320197 | MR 1155498 | Zbl 0746.65077
[3] Apel, Th., Heinrich, B.: Mesh refinement and windowing near edges for some elliptic problem. SIAM J. Numer. Anal. 31 (1994), 695-708. DOI 10.1137/0731037 | MR 1275108 | Zbl 0807.65122
[4] Apel, Th., Nicaise, S.: Elliptic problems in domains with edges: anisotropic regularity and anisotropic finite element meshes. Preprint TU Chemnitz-Zwickau SPC94\_16 (1994). MR 1399121
[5] Apel, Th., Nicaise, S.: Elliptic problems in domains with edges: anisotropic regularity and anisotropic finite element meshes. Partial Differential Equations and Functional Analysis. In Memory of Pierre Grisvard J. Cea, D. Chenais, G. Geymonat, J.-L. Lions Birkhäuser Boston (1996), 18-34 shortened version of Preprint SPC94\_16, TU Chemnitz-Zwickau, 1994. MR 1399121 | Zbl 0854.35005
[6] 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 (1996), 63-85. DOI 10.1002/(SICI)1099-1476(19960110)19:1<63::AID-MMA764>3.0.CO;2-S | MR 1365264
[7] Apel, Th., Sirch, D., Winkler, G.: Error estimates for control contstrained optimal control problems: Discretization with anisotropic finite element meshes. Preprint SPP1253-02-06 publ DFG Priority Program 1253 Erlangen (2008).
[8] Arada, N., Casas, E., Tröltzsch, F.: Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl. (2002), 23 201-229. DOI 10.1023/A:1020576801966 | MR 1937089 | Zbl 1033.65044
[9] Beagles, A. E., Whiteman, J. R.: Finite element treatment of boundary singularities by augmentation with non-exact singular functions. Numer. Methods Partial Differ. Equations 2 (1986), 113-121. DOI 10.1002/num.1690020203 | MR 0867853 | Zbl 0626.65112
[10] Casas, E., Mateos, M., Tröltzsch, F.: Error estimates for the numerical approximation of boundary semilinear elliptic control problems. Comput. Optim. Appl. 31 (2005), 193-219. DOI 10.1007/s10589-005-2180-2 | MR 2150243 | Zbl 1081.49023
[11] Clément, P.: Approximation by finite element functions using local regularization. Rev. Franc. Automat. Inform. Rech. Operat. 2 (1975), 77-84. MR 0400739
[12] Dauge, M.: Neumann and mixed problems on curvilinear polyhedra. Integral Equations Oper. Theory 15 (1992), 227-261. DOI 10.1007/BF01204238 | MR 1147281 | Zbl 0767.46026
[13] Falk, R. S.: Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl. 44 (1973), 28-47. DOI 10.1016/0022-247X(73)90022-X | MR 0686788 | Zbl 0268.49036
[14] Geveci, T.: On the approximation of the solution of an optimal control problem governed by an elliptic equation. RAIRO, Anal. Numér. 13 (1979), 313-328. MR 0555382 | Zbl 0426.65067
[15] Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics, Vol. 24. Pitman Boston (1985). MR 0775683
[16] Grisvard, P.: Singularities in boundary value problems. Research Notes in Applied Mathematics, Vol. 22. Masson/Springer Paris/Berlin (1992). MR 1173209
[17] Grisvard, P.: Singular behavior of elliptic problems in non Hilbertian Sobolev spaces. J. Math. Pures Appl., IX. Sér. 74 (1995), 3-33. MR 1313613 | Zbl 0854.35018
[18] Hinze, M.: A variational discretization concept in control constrained optimization: The linear-quadratic case. Comput. Optim. Appl. 30 (2005), 45-61. DOI 10.1007/s10589-005-4559-5 | MR 2122182 | Zbl 1074.65069
[19] Kondrat'ev, V. A.: Singularities of the solution of Dirichlet problem for a second-order elliptic equation in the neighborhood of an edge. Differ. Uravn. 13 (1977), 2026-2032 Russian. MR 0486987
[20] Kufner, A., Sändig, A.-M.: Some Applications of Weighted Sobolev Spaces. Teubner Leipzig (1987). MR 0926688
[21] Lubuma, J. M.-S., Nicaise, S.: Finite element method for elliptic problems with edge singularities. J. Comput. Appl. Math. 106 (1999), 145-168. DOI 10.1016/S0377-0427(99)00062-X | MR 1696808 | Zbl 0936.65130
[22] Malanowski, K.: Convergence of approximations vs. regularity of solutions for convex, control-constrained optimal-control problems. Appl. Math. Optimization 8 (1982), 69-95. DOI 10.1007/BF01447752 | MR 0646505 | Zbl 0479.49017
[23] Meyer, C., Rösch, A.: Superconvergence properties of optimal control problems. SIAM J. Control Optim. 43 (2004), 970-985. DOI 10.1137/S0363012903431608 | MR 2114385 | Zbl 1071.49023
[24] Michlin, S. G.: Partielle Differentialgleichungen in der mathematischen Physik. Akademie-Verlag Berlin (1978), German. MR 0513026 | Zbl 0397.35001
[25] Nazarov, S. A., Plamenevsky, B. A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. Walther de Gruyter & Co. Berlin (1994). MR 1283387 | Zbl 0806.35001
[26] mann, J. Roß: Gewichtete Sobolev-Slobodetskiĭ-Räume und Anwendungen auf elliptische Randwertaufgaben in Gebieten mit Kanten. Habilitationsschrift Universität Rostock Rostock (1988), German.
[27] mann, J. Roß: The asymptotics of the solutions of linear elliptic variational problems in domains with edges. Z. Anal. Anwend. 9 (1990), 565-578. MR 1119299
[28] Sändig, A.-M.: Error estimates for finite-element solutions of elliptic boundary value problems in non-smooth domains. Z. Anal. Anwend. 9 (1990), 133-153. MR 1063250
[29] Scott, L. R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990), 483-493. DOI 10.1090/S0025-5718-1990-1011446-7 | MR 1011446 | Zbl 0696.65007
[30] Zaionchkovskii, V., Solonnikov, V. A.: Neumann problem for second-order elliptic equations in domains with edges on the boundary. J. Sov. Math. 27 (1984), 2561-2586. DOI 10.1007/BF01103718
Partner of
EuDML logo