L 2 -error estimates for Dirichlet and Neumann problems on anisotropic finite element meshes

Thomas Apel; Dieter Sirch

Applications of Mathematics (2011)

  • Volume: 56, Issue: 2, page 177-206
  • ISSN: 0862-7940

Abstract

top
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.

How to cite

top

Apel, Thomas, and Sirch, Dieter. "$L^2$-error estimates for Dirichlet and Neumann problems on anisotropic finite element meshes." Applications of Mathematics 56.2 (2011): 177-206. <http://eudml.org/doc/116520>.

@article{Apel2011,
abstract = {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.},
author = {Apel, Thomas, Sirch, Dieter},
journal = {Applications of Mathematics},
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; Dirichlet and a Neumann boundary value problem; elliptic boundary value problem; a priori error estimate; interpolation of non-smooth functions; finite element error; non-convex domain; edge singularity; anisotropic mesh grading; Dirichlet and a Neumann boundary value problem},
language = {eng},
number = {2},
pages = {177-206},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$L^2$-error estimates for Dirichlet and Neumann problems on anisotropic finite element meshes},
url = {http://eudml.org/doc/116520},
volume = {56},
year = {2011},
}

TY - JOUR
AU - Apel, Thomas
AU - Sirch, Dieter
TI - $L^2$-error estimates for Dirichlet and Neumann problems on anisotropic finite element meshes
JO - Applications of Mathematics
PY - 2011
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 2
SP - 177
EP - 206
AB - 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.
LA - eng
KW - elliptic boundary value problem; a priori error estimates; interpolation of non-smooth functions; finite element error; non-convex domains; edge singularities; anisotropic mesh grading; Dirichlet and a Neumann boundary value problem; elliptic boundary value problem; a priori error estimate; interpolation of non-smooth functions; finite element error; non-convex domain; edge singularity; anisotropic mesh grading; Dirichlet and a Neumann boundary value problem
UR - http://eudml.org/doc/116520
ER -

References

top
  1. Apel, Th., 10.1051/m2an:1999139, M2AN, Math. Model. Numer. Anal. (1999), 33 1149-1185. (1999) Zbl0984.65113MR1736894DOI10.1051/m2an:1999139
  2. Apel, Th., Dobrowolski, M., 10.1007/BF02320197, Computing 47 (1992), 277-293. (1992) Zbl0746.65077MR1155498DOI10.1007/BF02320197
  3. Apel, Th., Heinrich, B., 10.1137/0731037, SIAM J. Numer. Anal. 31 (1994), 695-708. (1994) Zbl0807.65122MR1275108DOI10.1137/0731037
  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). (1994) MR1399121
  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. (1996) Zbl0854.35005MR1399121
  6. Apel, Th., Sändig, A.-M., Whiteman, J. R., 10.1002/(SICI)1099-1476(19960110)19:1<63::AID-MMA764>3.0.CO;2-S, Math. Methods Appl. Sci. 19 (1996), 63-85. (1996) MR1365264DOI10.1002/(SICI)1099-1476(19960110)19:1<63::AID-MMA764>3.0.CO;2-S
  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). (2008) 
  8. Arada, N., Casas, E., Tröltzsch, F., 10.1023/A:1020576801966, Comput. Optim. Appl. (2002), 23 201-229. (2002) Zbl1033.65044MR1937089DOI10.1023/A:1020576801966
  9. Beagles, A. E., Whiteman, J. R., 10.1002/num.1690020203, Numer. Methods Partial Differ. Equations 2 (1986), 113-121. (1986) Zbl0626.65112MR0867853DOI10.1002/num.1690020203
  10. Casas, E., Mateos, M., Tröltzsch, F., 10.1007/s10589-005-2180-2, Comput. Optim. Appl. 31 (2005), 193-219. (2005) Zbl1081.49023MR2150243DOI10.1007/s10589-005-2180-2
  11. Clément, P., Approximation by finite element functions using local regularization, Rev. Franc. Automat. Inform. Rech. Operat. 2 (1975), 77-84. (1975) MR0400739
  12. Dauge, M., 10.1007/BF01204238, Integral Equations Oper. Theory 15 (1992), 227-261. (1992) Zbl0767.46026MR1147281DOI10.1007/BF01204238
  13. Falk, R. S., 10.1016/0022-247X(73)90022-X, J. Math. Anal. Appl. 44 (1973), 28-47. (1973) Zbl0268.49036MR0686788DOI10.1016/0022-247X(73)90022-X
  14. Geveci, T., 10.1051/m2an/1979130403131, RAIRO, Anal. Numér. 13 (1979), 313-328. (1979) Zbl0426.65067MR0555382DOI10.1051/m2an/1979130403131
  15. Grisvard, P., Elliptic Problems in Nonsmooth Domains. Monographs and Studies in Mathematics, Vol. 24, Pitman Boston (1985). (1985) MR0775683
  16. Grisvard, P., Singularities in boundary value problems. Research Notes in Applied Mathematics, Vol. 22, Masson/Springer Paris/Berlin (1992). (1992) MR1173209
  17. Grisvard, P., Singular behavior of elliptic problems in non Hilbertian Sobolev spaces, J. Math. Pures Appl., IX. Sér. 74 (1995), 3-33. (1995) Zbl0854.35018MR1313613
  18. Hinze, M., 10.1007/s10589-005-4559-5, Comput. Optim. Appl. 30 (2005), 45-61. (2005) Zbl1074.65069MR2122182DOI10.1007/s10589-005-4559-5
  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. (1977) MR0486987
  20. Kufner, A., Sändig, A.-M., Some Applications of Weighted Sobolev Spaces, Teubner Leipzig (1987). (1987) MR0926688
  21. Lubuma, J. M.-S., Nicaise, S., 10.1016/S0377-0427(99)00062-X, J. Comput. Appl. Math. 106 (1999), 145-168. (1999) Zbl0936.65130MR1696808DOI10.1016/S0377-0427(99)00062-X
  22. Malanowski, K., 10.1007/BF01447752, Appl. Math. Optimization 8 (1982), 69-95. (1982) Zbl0479.49017MR0646505DOI10.1007/BF01447752
  23. Meyer, C., Rösch, A., 10.1137/S0363012903431608, SIAM J. Control Optim. 43 (2004), 970-985. (2004) Zbl1071.49023MR2114385DOI10.1137/S0363012903431608
  24. Michlin, S. G., Partielle Differentialgleichungen in der mathematischen Physik, Akademie-Verlag Berlin (1978), German. (1978) Zbl0397.35001MR0513026
  25. Nazarov, S. A., Plamenevsky, B. A., Elliptic Problems in Domains with Piecewise Smooth Boundaries, Walther de Gruyter &amp; Co. Berlin (1994). (1994) Zbl0806.35001MR1283387
  26. mann, J. Roß, Gewichtete Sobolev-Slobodetskiĭ-Räume und Anwendungen auf elliptische Randwertaufgaben in Gebieten mit Kanten, Habilitationsschrift Universität Rostock Rostock (1988), German. (1988) 
  27. mann, J. Roß, 10.4171/ZAA/424, Z. Anal. Anwend. 9 (1990), 565-578. (1990) MR1119299DOI10.4171/ZAA/424
  28. Sändig, A.-M., 10.4171/ZAA/388, Z. Anal. Anwend. 9 (1990), 133-153. (1990) MR1063250DOI10.4171/ZAA/388
  29. Scott, L. R., Zhang, S., 10.1090/S0025-5718-1990-1011446-7, Math. Comput. 54 (1990), 483-493. (1990) Zbl0696.65007MR1011446DOI10.1090/S0025-5718-1990-1011446-7
  30. Zaionchkovskii, V., Solonnikov, V. A., 10.1007/BF01103718, J. Sov. Math. 27 (1984), 2561-2586. (1984) DOI10.1007/BF01103718

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.