Mixed Finite Element Methods for Nonlinear Second-Order Elliptic Problems

Park, Eun Jae

doi:10.1137/0732040

Cited by 85 publications

(48 citation statements)

References 16 publications

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“…In particular, we prove that Newton's method converges quadratically, but not uniformly. This confirms the convergence analysis for nonlinear second order elliptic problems studied by Douglas and Dupont in [15] and by Park in [29].…”

Section: Position Of the Papersupporting

confidence: 76%

Finite element discretization of Darcy's equations with pressure dependent porosity

Girault¹,

Murat²,

Salgado³

2010

ESAIM: M2AN

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Abstract.We consider the flow of a viscous incompressible fluid through a rigid homogeneous porous medium. The permeability of the medium depends on the pressure, so that the model is nonlinear. We propose a finite element discretization of this problem and, in the case where the dependence on the pressure is bounded from above and below, we prove its convergence to the solution and propose an algorithm to solve the discrete system. In the case where the dependence on the pressure is exponential, we propose a splitting scheme which involves solving two linear systems, but parts of the analysis of this method are still heuristic. Numerical tests are presented, which illustrate the introduced methods.Mathematics Subject Classification. 76S05, 65N30.

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Section: Position Of the Papersupporting

confidence: 76%

Finite element discretization of Darcy's equations with pressure dependent porosity

Girault¹,

Murat²,

Salgado³

2010

ESAIM: M2AN

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“…The Newton method, which is quadratically convergent was very successfully applied to Richards' equation in e.g. [7,8,20,23,27]. The drawback of Newton's method is that it is only locally convergent and involves the computation of derivatives.…”

Section: ∂ T θ(ψ ) − ∇ · (K(θ(ψ ))∇(ψmentioning

confidence: 99%

A study on iterative methods for solving Richards’ equation

2016

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This work concerns linearization methods for efficiently solving the Richards equation, a degenerate elliptic-parabolic equation which models flow in saturated/unsaturated porous media. The discretization of Richards' equation is based on backward Euler in time and Galerkin finite elements in space. The most valuable linearization schemes for Richards' equation, i.e. the Newton method, the Picard method, the Picard/Newton method and the L−scheme are presented and their performance is comparatively studied. The convergence, the computational time and the condition numbers for the underlying linear systems are recorded. The convergence of the L−scheme is theoretically proved and the convergence of the other methods is discussed. A new scheme is proposed, the L−scheme/Newton method which is more robust and quadratically convergent. The linearization methods are tested on illustrative numerical examples.

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“…Hence K |f (ũ n H )| 2 k+1,2,K is bounded by a constant C which is independent of H. Since α(x) > α 0 > 0, (24) …”

Section: Error Analysismentioning

confidence: 99%

“…Linear and nonlinear second order elliptic problems are studied in [25,11,3,26,23,24,19,20]. Two-scale mixed methods for parabolic problems are studied in references [10,30,6].…”

Section: Introductionmentioning

confidence: 99%

Two-Scale Product Approximation for Semilinear Parabolic Problems in Mixed Methods

Kim

Park²,

Seo

2014

Journal of the Korean Mathematical Society

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Abstract. We propose and analyze two-scale product approximation for semilinear heat equations in the mixed finite element method. In order to efficiently resolve nonlinear algebraic equations resulting from the mixed method for semilinear parabolic problems, we treat the nonlinear terms using some interpolation operator and exploit a two-scale grid algorithm. With this scheme, the nonlinear problem is reduced to a linear problem on a fine scale mesh without losing overall accuracy of the final system. We derive optimal order L ∞ ((0, T ]; L 2 (Ω))-error estimates for the relevant variables. Numerical results are presented to support the theory developed in this paper.

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Mixed Finite Element Methods for Nonlinear Second-Order Elliptic Problems

Cited by 85 publications

References 16 publications

Finite element discretization of Darcy's equations with pressure dependent porosity

Finite element discretization of Darcy's equations with pressure dependent porosity

A study on iterative methods for solving Richards’ equation

Two-Scale Product Approximation for Semilinear Parabolic Problems in Mixed Methods

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