A note on the approximation of free boundaries by finite element methods

Nochetto, Ricardo H.

doi:10.1051/m2an/1986200203551

Cited by 38 publications

(32 citation statements)

References 25 publications

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“…The usual technique to approximate (1.1) (with or without regularization of 0) amounts to discretizing a nonlinear elliptic partial differential equation at each time step. Both, theoretical and numerical results, are now well known for these schemes [10], [23], [16], [17], [18], [8], [20], [25] which behave rather well in approximating not only solutions, but also interfaces [19], [20]. However, when dealing with nonlinear problems, one usually tries to linearize them so as to take advantage of efficient linear solvers [7], [15].…”

Section: Introductionmentioning

confidence: 99%

An Efficient Linear Scheme to Approximate Parabolic Free Boundary Problems: Error Estimates and Implementation

Nochetto¹,

Verdi²

1988

Mathematics of Computation

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Abstract.This paper deals with a fully discrete scheme to approximate multidimensional singular parabolic problems; two-phase Stefan problems and porous medium equations are included. The algorithm consists of approximating at each time step a linear elliptic partial differential equation by piecewise linear finite elements and then making an element-by-element algebraic correction to account for the nonlinearity. Several energy error estimates are derived for the physical unknowns; a sharp rate of convergence of 0(h1/2) is our main result. The crucial point in implementing the scheme is the efficient resolution of linear systems involved. This topic is discussed, and the results of several numerical experiments are shown. Introduction.The aim of this paper is to analyze from both a theoretical and computational viewpoint the performance of a linear scheme to approximate the following multidimensional parabolic problem:

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Section: Introductionmentioning

confidence: 99%

An Efficient Linear Scheme to Approximate Parabolic Free Boundary Problems: Error Estimates and Implementation

Nochetto¹,

Verdi²

1988

Mathematics of Computation

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“…We require, for physical coherence, that X (S) is strictly increasing for 0 < S < 1. By using (33) and (34),…”

Section: Mathematical Model When Capillary Effects Are Non-negligiblementioning

confidence: 99%

“…We use mass lumping approach [33]- [34] with weakly-acute mesh to garanty the monotonicity. For the diffusion part we use the implicit scheme and for the non-linear term in the time derivative the explicit scheme.…”

Section: Discrete Scheme Used In the Mmodd-scilab Implementationmentioning

confidence: 99%

Isothermal water flows in low porosity porous media in presence of vapor–liquid phase change

Farína¹,

Bodin

Clopeau

et al. 2014

Nonlinear Analysis: Real World Applications

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In this article we consider a mathematical model for a low porosity porous medium saturated by water, present both as the liquid and the vapor phase. In the isothermal case we propose a new formulation using a single nonlinear parabolic-hyperbolic equation for the fluid mixture density X. We present the derivation of the unified model and a number of numerical simulations based on regularization and Kirchhoff's transform.

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“…Hence there appears to be no gain in accuracy from regularizing. In addition, one can apply the techniques of Nochetto (1986Nochetto ( , 1988 to recover an approximation F h to the free boundary F, see below. Once again there appears to be no advantage from regularizing.…”

Section: Proofmentioning

confidence: 99%

Finite element approximation of a model reaction ? diffusion problem with a non-Lipschitz nonlinearity

Barrett

Shanahan

1991

Numer. Math.

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Summary. Given 2>0 and p~(0, 1), we consider the following problem: findwhere ~r~c~-~2 is a smooth convex domain. We prove optimal H 1 and L ~ error bounds for the standard continuous piecewise linear Galerkin finite element approximation. In addition we analyse a more practical approximation using numerical integration on the nonlinear term. Finally we consider a modified nonlinear SOR algorithm, which is shown to be globally convergent, for solving the algebraic system derived from the more practical approximation.

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A note on the approximation of free boundaries by finite element methods

Cited by 38 publications

References 25 publications

An Efficient Linear Scheme to Approximate Parabolic Free Boundary Problems: Error Estimates and Implementation

An Efficient Linear Scheme to Approximate Parabolic Free Boundary Problems: Error Estimates and Implementation

Isothermal water flows in low porosity porous media in presence of vapor–liquid phase change

Finite element approximation of a model reaction ? diffusion problem with a non-Lipschitz nonlinearity

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