Existence of solutions for the time-dependent thermistor equations

Allegretto, W.; Xie, Hong

doi:10.1093/imamat/48.3.271

Cited by 42 publications

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“…The existence of global weak solutions in two space dimensions (il c R2) was shown by Cimatti [5], see also Rodrigues [15] and Allegretto and Xie [2] for existence results for related problems. The regularity of these solutions, however, is insufficient for the purpose of proving error bounds of optimal order, cf.…”

Section: Existence and Regularitymentioning

confidence: 86%

“…In particular, we avoid using a maximum norm bound for V</>/,, which would be difficult to obtain. We also consider a completely discrete scheme based on the backward Euler method with semi-implicit linearization: find Un £ Sh , ®n € Sh with <!>" -nhg(tn) £ Sh such that (BnUn , X) + (Vt/", V*) = (o(Un-X)\V<!>n-X\2, X), (1)(2)(3)(4)(5)(6)(7)(8) Vxe$k, tn€(0,T], Uo = Uho , and (1.9) (cr(c7n)V<D",Vy) = 0, V* £ Sh, tn £ [0, T).…”

Section: Introductionmentioning

confidence: 99%

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A Finite Element Model for the Time-Dependent Joule Heating Problem

Elliott¹,

Larsson²

1995

Mathematics of Computation

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Abstract. We study a spatially semidiscrete and a completely discrete finite element model for a nonlinear system consisting of an elliptic and a parabolic partial differential equation describing the electric heating of a conducting body. We prove error bounds of optimal order under minimal regularity assumptions when the number of spatial variables d < 3. We establish the existence of solutions with the required regularity over arbitrarily long intervals of time when d < 2 .

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Section: Existence and Regularitymentioning

confidence: 86%

Section: Introductionmentioning

confidence: 99%

A Finite Element Model for the Time-Dependent Joule Heating Problem

Elliott¹,

Larsson²

1995

Mathematics of Computation

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“…The unknown u is the temperature, φ is the electric potential, σ (u) is the temperature-dependent electric conductivity and σ (u)|∇φ| 2 is the source of Joule heating and the initial u 0 (X) and g(X, t) are given smooth functions.…”

Section: Introductionmentioning

confidence: 99%

“…A lot of studies of theoretical and numerical analysis have been devoted to system (1.1)-(1.4) (see e.g., [1][2][3][4][5][6][7][8][9][10][11][12] and references therein). More precisely, the existence of solutions was showed in [2].…”

Section: Introductionmentioning

confidence: 99%

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Superconvergent estimates of conforming finite element method for nonlinear time‐dependent Joule heating equations

Shi

Yang

2017

Numerical Methods Partial

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In this article, we study the superconvergence analysis of conforming bilinear finite element method (FEM) for nonlinear Joule heating equations. Based on the rigorous estimates together with high accuracy analysis of this element, mean value technique and interpolation postprocessing approach, the superclose and superconvergent estimates about the related variables in H 1 -norm are derived for semidiscrete and a linearized backward Euler fully discrete schemes, which extends the results of optimal estimates obtained for conforming FEMs in the previous literature. At last, a numerical experiment is performed to verify the theoretical analysis. K E Y W O R D SNonlinear Joule heating equations, bilinear FEM, semidiscrete and linearized fully discrete schemes, supercloseness and superconvergence analysis

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The theta‐Galerkin finite element method for coupled systems resulting from microsensor thermistor problems

Mbehou

2017

Math Methods in App Sciences

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This paper is devoted to the analysis of a linearized theta‐Galerkin finite element method for the time‐dependent coupled systems resulting from microsensor thermistor problems. Hereby, we focus on time discretization based on θ‐time stepping scheme with θ∈false[12,1false) including the standard Crank‐Nicolson ( θ=12) and the shifted Crank‐Nicolson ( θ=12+δ, where δ is the time‐step) schemes. The semidiscrete formulation in space is presented and optimal error bounds in L2‐norm and the energy norm are established. For the fully discrete system, the optimal error estimates are derived for the standard Crank‐Nicolson, the shifted Crank‐Nicolson, and the general case where θ≠12+kδ with k=0,1 . Finally, numerical simulations that validate the theoretical findings are exhibited.

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Existence of solutions for the time-dependent thermistor equations

Cited by 42 publications

References 0 publications

A Finite Element Model for the Time-Dependent Joule Heating Problem

A Finite Element Model for the Time-Dependent Joule Heating Problem

Superconvergent estimates of conforming finite element method for nonlinear time‐dependent Joule heating equations

The theta‐Galerkin finite element method for coupled systems resulting from microsensor thermistor problems

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