On Galerkin Approximations of a Quasilinear Nonpotential Elliptic Problem of a Nonmonotone Type

Hlaváček, Ivan; Křížek, Michal; Malý, Jan

doi:10.1006/jmaa.1994.1192

Cited by 72 publications

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“…To conclude this section, we would like to point out that after this paper had been submitted for publication, it came to our attention that a similar (uniqueness) result had been given in [4]. However, there is no further discussion in [4] as we do in Section 4.…”

mentioning

confidence: 76%

Uniqueness of weak solution for nonlinear elliptic equations in divergence form

Zhang¹

2000

International Journal of Mathematics and Mathematical Sciences

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confidence: 76%

Uniqueness of weak solution for nonlinear elliptic equations in divergence form

Zhang¹

2000

International Journal of Mathematics and Mathematical Sciences

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“…11.6] for a short proof of the uniqueness of the solution. In [22], the existence and the uniqueness of a weak solution of Problem (1) are shown for f ∈ L 2 (Ω ), with more general mixed Dirichlet-Neumann boundary conditions, on a bounded domain with a Lipschitz boundary. For the proof of the uniqueness, the divergence form of the differential operator is an essential ingredient.…”

Section: Remarkmentioning

confidence: 99%

A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems

Abdulle

Vilmart

2011

Numer. Math.

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To cite this version:Assyr Abdulle, Gilles Vilmart. A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems. Numerische Mathematik, Springer Verlag, 2012, 121, pp.397-431 Abstract The effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied. Under similar assumption on the quadrature formula as for linear problems, optimal error estimates in the L 2 and the H 1 norms are proved. The numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficiently fine mesh. The analysis is valid for both simplicial and rectangular finite elements of arbitrary order. Numerical experiments corroborate the theoretical convergence rates.Keywords nonmonotone quasilinear elliptic problem · a priori error estimates · numerical quadrature · variational crime · finite elements.

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“…There are just a few uniqueness results in the case of sufficiently small data u, Brenner and Scott [2], pp. 188-191, or when the discretization parameter h is large enough, Hlaváček [9] and Hlaváček et al [10]. Moreover these papers assume the coefficient a(x, y) of the quasilinear equation to be bounded on Ω × R. In this case, we are able to prove the uniqueness of the solution of the discrete equation for every h small enough for any u ∈ L p (Ω) when p > n; see Corollary 3.3.…”

Section: Introductionmentioning

confidence: 96%

Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations

Casas¹,

Tröltzsch²

2010

ESAIM: COCV

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Abstract.In this paper, we carry out the numerical analysis of a distributed optimal control problem governed by a quasilinear elliptic equation of non-monotone type. The goal is to prove the strong convergence of the discretization of the problem by finite elements. The main issue is to get error estimates for the discretization of the state equation. One of the difficulties in this analysis is that, in spite of the partial differential equation has a unique solution for any given control, the uniqueness of a solution for the discrete equation is an open problem.Mathematics Subject Classification. 49M25, 35J60, 35B37, 65N30.

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On Galerkin Approximations of a Quasilinear Nonpotential Elliptic Problem of a Nonmonotone Type

Cited by 72 publications

References 0 publications

Uniqueness of weak solution for nonlinear elliptic equations in divergence form

Uniqueness of weak solution for nonlinear elliptic equations in divergence form

A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems

Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations

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