Transformation of dependent variables and the finite element solution of nonlinear evolution equations

Čermák, Libor; Zlámal, Miloš

doi:10.1002/nme.1620150104

Cited by 16 publications

(2 citation statements)

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“…This allows, in particular, to use the Newton method for the solution of the arising system of nonlinear algebraic equations, albeit its use without regularization has been advocated in Wheeler [52] or in Baughman and Walkington [4] and studied in Kelley and Rulla [28]. Alternative approaches such as transformation of dependent variables ofČermák and Zlámal [10] have also been proposed.…”

Section: Introductionmentioning

confidence: 99%

Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem

2014

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We consider in this paper the time-dependent two-phase Stefan problem and derive a posteriori error estimates and adaptive strategies for its conforming spatial and backward Euler temporal discretizations. Regularization of the enthalpy-temperature function and iterative linearization of the arising systems of nonlinear algebraic equations are considered. Our estimators yield a guaranteed and fully computable upper bound on the dual norm of the residual, as well as on the L 2 (L 2 ) error of the temperature and the L 2 (H −1 ) error of the enthalpy. Moreover, they allow to distinguish the space, time, regularization, and linearization error components. An adaptive algorithm is proposed, which ensures computational savings through the online choice of a sufficient regularization parameter, a stopping criterion for the linearization iterations, local space mesh refinement, time step adjustment, and equilibration of the spatial and temporal errors. We also prove the efficiency of our estimate. Our analysis is quite general and is not focused on a specific choice of the space discretization and of the linearization. As an example, we apply it to the vertex-centered finite volume (finite element with mass lumping and quadrature) and Newton methods. Numerical results illustrate the effectiveness of our estimates and the performance of the adaptive algorithm.MSC: 65N08, 65N15, 65N50, 80A22 IntroductionThe two-phase Stefan problem models a phase change process which is governed by the Fourier law, c.f. Friedman [22]. The two phases, typically solid and liquid, are separated by a moving interface, whose motion is governed by the so-called Stefan condition., be an open bounded polygonal or polyhedral domain, not necessarily convex, and let T > 0. The mathematical statement of the problem is as follows: given an initial enthalpy u 0 and a source function f , find the enthalpy u such that(1.1c) * This work was supported by the ERT project "Enhanced oil recovery and geological sequestration of CO 2 : mesh adaptivity, a posteriori error control, and other advanced techniques" (LJLL/IFPEN) † daniele.di-pietro@univ-montp2.fr ‡

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Section: Introductionmentioning

confidence: 99%

Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem

2014

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“…The well-known Kirchhoff transformation (see [2,9,17]), which changes the nonlinear problem to a linear one, can be applied in the case of isotropic nonlinear media, i.e., when A is a scalar function. However, it cannot be applied to prove the existence of u in the case of anisotropic nonlinear media, in general.…”

Section: Introductionmentioning

confidence: 99%

On a comparison principle for a quasilinear elliptic boundary value problem of a nonmonotone type

Křížek¹,

Liu²

1996

Appl. Math. (Warsaw)

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Interpolation of coefficients and transformation of the dependent variable in finite element methods for the non‐linear heat equation

Larsson

Thomée

Zhang

et al. 1989

Math Methods in App Sciences

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Transformation of dependent variables and the finite element solution of nonlinear evolution equations

Cited by 16 publications

References 7 publications

Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem

Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem

On a comparison principle for a quasilinear elliptic boundary value problem of a nonmonotone type

Interpolation of coefficients and transformation of the dependent variable in finite element methods for the non‐linear heat equation

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