# Efficient numerical solution of Neumann problems on complicated domains

## Abstract

We consider elliptic partial differential equations with Neumann boundary conditions on complicated domains. The discretization is performed by composite finite elements.

The a priori error analysis typically is based on precise knowledge of the regularity of the solution. However, the constants in the regularity estimates possibly depend critically on the geometric details of the domain and the analysis of their quantitative influence is rather involved.

Here, we consider a polyhedral Lipschitz domain Ω with a possibly huge number of geometric details ranging from size O(ε) to O(1). We assume that Ω is a perturbation of a simpler Lipschitz domain Ω. We prove error estimates where only the regularity of the partial differential equation on Ω is needed along with bounds on the norm of extension operators which are explicit in appropriate geometric parameters.

Since composite finite elements allow a multiscale discretization of problems on complicated domains, the linear system which arises can be solved by a simple multi-grid method. We show that this method converges at an optimal rate independent of the geometric structure of the problem.

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## References

1. 1. Bank, R., Smith, R.: An algebraic multilevel multigraph algorithm. SIAM J. Sci. Comput. 23, 1572–1592 (2002)

2. 2. Bank, R., Xu, J.: A hierarchical basis multigrid method for unstructured grids. In: Hackbusch, W., Wittum, G. (eds.): Fast solvers for flow problems. Braunschweig: Vieweg 1995, pp. 1–13

3. 3. Bank, R., Xu, J.: An algorithm for coarsening unstructured meshes. Numer. Math. 73, 1–36 (1996)

4. 4. Braess, D.: Towards algebraic multigrid for elliptic problems of second order. Computing 55, 379–393 (1995)

5. 5. Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. New York: Springer 1994

6. 6. Chan, T.F., Smith, B.F.: Domain decomposition and multi-grid algorithms for elliptic problems on unstructured meshes. Electron. Trans. Numer. Anal. 2, 171–182 (1994)

7. 7. Chan, T.F., Xu, J., Zikatanov, L.: An agglomeration multigrid method for unstructured grids. In: Mandel, J. et al. (eds.): Domain decomposition methods. 10 (Contemporary. Mathematics 218) Providence, RI: AMS 1998, pp. 67–81 67–81

8. 8. Feuchter, D., Heppner, I., Sauter, S., Wittum, G.: Bridging the gap between geometric and algebraic multi-grid methods. Comput. Vis. Sci. 6, 1–13 (2003)

9. 9. Griebel, M., Knapek, S.: A multigrid-homogenization method. In: Helmig, R. et al. (eds.): Modeling and computation in environmental sciences. (Notes on Numerical Fluid Mechanics 59) Braunschweig: Vieweg 1997, pp. 187–202

10. 10. Grisvard, P.: Elliptic problems in nonsmooth domains. (Monographs and Studies in Mathematics 21) Boston: Pitman 1985

11. 11. Hackbusch, W.: Multigrid methods and applications. Berlin: Springer 1985 (2nd edition 2003)

12. 12. Hackbusch, W.: Elliptic differential equations. Theory and numerical treatment. (Springer Series in Computational Mathematics 18) Berlin: Springer 1992

13. 13. Hackbusch, W., Sauter, S.: Composite finite elements for problems containing small geometric details. II. Implementation and numerical results. Comput. Vis. Sci. 1, 15–25 (1997)

14. 14. Kornhuber, R., Yserentant, H.: Multilevel methods for elliptic problems on domains not resolved by the coarse grid. In: Keyes, D.E., Xu, J. (eds.): Domain decomposition methods in scientific and engineering computing. (Contemporay Mathematics 180) Providence, RI: AMS 1994, pp. 49–60

15. 15. Mandel, J., Brezina, M., Vaněk, P.: Energy optimization of algebraic multigrid bases. Computing 62, 205–228 (1999)

16. 16. Maz'ja, V.G.: Sobolev spaces. Berlin: Springer 1985

17. 17. McLean, W.: Strongly elliptic systems and boundary integral equations. Cambridge: Cambridge Univ. Press 2000

18. 18. Oleĭnik, O., Shamaev, A., Yosifian, G.: Mathematical problems in elasticity and homogenization. Amsterdam: North-Holland 1992

19. 19. Ruge, J., Stüben, K.: Algebraic multigrid. In: McCormick, S. (ed.): Multigrid methods. Philadelphia: SIAM 1987, pp. 73–130

20. 20. Sauter, S.A., Warnke, R.: Extension operators and approximation on domains containing small geometric details. East-West J. Numer. Math. 7, 61–77 (1999)

21. 21. Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54, 483–493 (1990)

22. 22. Scott, L.R., Zhang, S.: Higher dimensional nonnested multigrid methods. Math. Comp. 58, 457–466 (1992)

23. 23. Stahn, N.: Composite finite elements and multi-grid. Dissertation. Zürich: Institut für Mathematik, Universität Zürich 2006

24. 24. Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton: Princeton Univ. Press 1970

25. 25. Vaněk, P., Mandel, J., Brezina, M.: Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing 56, 179–196 (1996)

26. 26. Warnke, R.: Fortsetzungsoperatoren auf perforierten Gebieten. Diplomarbeit. Kiel: Mathematisches Seminar, Universität Kiel 1997

27. 27. Xu, J.: The auxiliary space method and optimal multigrid preconditioning techniques for unstructured grids. Computing 56, 215–235 (1996)

28. 28. Yserentant, H.: Coarse grid spaces for domains with a complicated boundary. Numer. Algorithms 21, 387–392 (1999)

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