# ERROR ESTIMATION FOR NONLINEAR ELLIPTIC PROBLEMS USING THE h-p-MIXED FINITE ELEMENT METHOD IN 3 DIMENSIONAL SPACE

• Published : 2001.05.01

#### Abstract

The approximation properties for $L^2$-projection, Raviart-Thomas projection, and inverse inequality have been derived in 3 dimensional space. h-p-mixed finite element methods for strongly nonlinear second order elliptic problems are proposed and analyzed in 3D. Solvability and convergence of the linearized problem have been shown through duality argument and fixed point argument. The analysis is carried out in detail using Raviart-Thomas-Nedelec spaces as an example.

#### References

1. Sobolev Spaces R. A. Adams
2. Numer. Math v.37 Error estimates for the combined h and p versions of finite element method I. Babuska;M. Dorr
3. Comput. Mechanics v.1 The h-p version of the finite element method Part 2: General results and applications B. Guo;I. Babuska
4. RAIRO Model. Math. Anal. Numer v.21 The h-p version of the finite element method with quasiuniform meshes I. Babuska;M. Suri
5. SIAM J. Num. Anal v.18 The p=version of finite element method I. Babuska;B. Szabo;I. Katz
6. Springer-Verlag Interpolation Spaces: An Introductions J. Bergh;J. Lofstrom
7. RAIRO, Anal. Numer v.2 In the existence, uniqueness, and approximation of saddle point problems arising from Lagrangian multipliers F. Brezzi
8. Math. Comput. v.38 Approximation results for orthogonal polynomials in Sobolev spaces C. Canuto;A. Quateroni
9. Center for Applied Mathematics One the Existence, Uniqueness and Convergence of Nonlinear Mixed Finite Element Methods Z. Chen
10. The finite element method for elliptic problems P. G. Ciarlet
11. Math. Comp v.29 A Galerkin method for a nonlinear Dirichlet problem J. Douglas Jr.;T. Dupont
12. Math. Comp. v.44 Global estimates for mixed methods for second order elliptic equations J. Douglas Jr.;J. E. Roberts
13. 2nd ed. Springer-Verlag Elliptic Partial Differential Equations of Second Order. D. Gilbarg;N. S. Trudinger
14. RIM-GARC Preprint Series An elliptic regularity of a Helmholtz-Type problem with an absorbing boungary condition J. Kim;D. Sheen
15. Num. Meth. Partial Different Equ. v.12 Mixed finite element method for nonlinear elliptic problems: The p-version M. Y. Lee.;F. A. Milner
16. J. Comp. Appl. Math. v.85 Mixed finite element method for nonlinear elliptic problems: The h-p version
17. Math. Comp. v.44 Mixed finite element methods for quasilinear second-order elliptic problems F. A. Milner
18. M²An v.26 Mixed finite element methods for Quasilinear second order elliptic problems: the p-bersion F. A. Milner;M. Suri.
19. Numer. Math. v.50 A new family of mixed finite elements in R³ J. C. Nedelec
20. Numer. Math v.35 Mixed Finite Elements in R³
21. to appear in SIAM J. Num. Anal. Mixed finite element methods for nonlinear second order elliptic problems E. J. Park
22. Proceed. Conf. on Mathematical Aspects of Finite Element Methods 606 of Lecture Notes in Mathematics mixed finite element method for 2nd order elliptic problems P. A. Raviart;J. M. Thomas;G. F. Hewitt, J. M. Delhaye, N. Zuber(eds.)
23. Math. Comp. v.54 On the stability and convergence of higher order mixed finite element methods for second order elliptic problems M. Suri.
24. The Theory of Approximation of Functions of Real Variable A. F. Timan
25. Interpolation Theory, Function Spaces, Differential Opernators H. Triebel