Expanded mixed finite element methods for quasilinear second order elliptic problems. II.(English)Zbl 0910.65080
This is the second paper of a series in which the author presents a new mixed formulation for the numerical solution of second-order elliptic problems [for part I see ibid. 32, No. 4, 479-499 (1998; reviewed above)]. This new formulation expands the standard mixed formulation in the sense that three variables are explicitly treated: the scalar unknown, its gradient, and its flux (the coefficient times the gradient). Based on this formulation, mixed finite element approximations of the quasilinear second-order elliptic problems are considered. Existence and uniqueness of the solution of the mixed formulation and its discretization are demonstrated. Optimal order error estimates in the \(L^p\)- and \(H^{-s}\)-norms are obtained for the mixed approximations. A postprocessing method for improving the scalar variable is analyzed, and superconvergent estimates are derived. Implementation techniques for solving systems of algebraic equations are discussed. Comparisons between the standard and expanded mixed formulations are given both theorically and experimentally. The mixed formulation proposed here is suitable for the case where the coefficient of differential equations is a small tensor and does not need to be inverted.
Reviewer: K.Najzar (Praha)
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
Keywords:
expanded mixed finite element method;error estimate;superconvergence;quasilinear second-order elliptic problem;postprocessingCitations:
Zbl 0910.65079Cite
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