Global Superconvergence and A Posteriori Error Estimators of the Finite Element Method for a Quasi-linear Elliptic Boundary Value Problem of Nonmonotone Type

Liu, Liping; Liu, Tang; Křížek, Michal; Lin, Tao; Zhang, Shuhua

doi:10.1137/s0036142903428402

Cited by 11 publications

(4 citation statements)

References 14 publications

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“…Demlow considered a more general case which included

a (u) = \frac{1}{\sqrt{1 + | \nabla u |^{2}}}

and presented two types of pointwise a posteriori error estimates for gradient error in piecewise linear finite element approximation. Recently, using the superconvergent approximation properties of finite element solutions, postprocessing‐based a posteriori error estimates for quasi‐linear elliptic problems have been proposed in . The residual‐based a posteriori error estimates of h ‐version of the discontinuous Galerkin method for have been studied in .…”

Section: Introductionmentioning

confidence: 99%

“…The a posteriori error estimates of the finite element method for linear elliptic problems have been widely studied, see for instance and the references therein. The a posteriori error estimates of the finite element method for nonlinear elliptic problems have been considered in . In , Verfürth gave a general framework of a posteriori error estimates in H 1 ‐norm and L 2 ‐norm for nonlinear problems, respectively.…”

Section: Introductionmentioning

confidence: 99%

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A posteriori error estimates of edge residual type of finite element method for nonmonotone quasi‐linear elliptic problems

Guo

Huang

Wang

2013

Numerical Methods Partial

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In this article, we study the edge residual-based a posteriori error estimates of conforming linear finite element method for nonmonotone quasi-linear elliptic problems. It is proven that edge residuals dominate a posteriori error estimates. Up to higher order perturbations, edge residuals can act as a posteriori error estimators. The global reliability and local efficiency bounds are established both in H 1 -norm and L 2 -norm. Numerical experiments are provided to illustrate the performance of the proposed error estimators.

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“…Demlow considered a more general case which included

a (u) = \frac{1}{\sqrt{1 + | \nabla u |^{2}}}

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A posteriori error estimates of edge residual type of finite element method for nonmonotone quasi‐linear elliptic problems

Guo

Huang

Wang

2013

Numerical Methods Partial

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“…The study for superconvergence of finite element methods is a topic of importance. See, for example, [1][2][3][4][5][6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Global Superconvergence Estimates of Petrov-Galerkin Methods for Hyperbolic Equations

Yang

2010

2010 2nd International Workshop on Database Technology and Applications

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In this paper, Petrov-Galerkin methods are employed for numerical solutions to hyperbolic partial differential equations, which are important mathematical models with many applications in engineering, such as wave transmission, petroleum reservoir. By virtue of superclose analysis and an interpolation postprocessing technique, the global superconvergence is obtained, which is new even for elliptic partial differential equations.

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“…The supercloseness property, which follows from the supraconvergence results, plays an important rôle in postprocessing, recovery techniques for the gradient, and a posteriori error estimates (see [1,2,8,9,[16][17][18][19]29,30,35,36,38,52] and the references cited therein). In all these papers, either a uniform grid or a smooth transformation of a uniform grid (cf., e.g., [35]) has to be assumed in order to obtain optimal second-order convergence for the gradient in the L 2 (Ω)-norm if u ∈ H 3 (Ω).…”

Section: Introductionmentioning

confidence: 99%

Supraconvergence and Supercloseness of a Discretisation for Elliptic Third-kind Boundary-value Problems on Polygonal Domains

Emmrich

2007

Comput. Methods Appl. Math.

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The third-kind boundary-value problem for a second-order elliptic equation on a polygonal domain with variable coefficients, mixed derivatives, and first-order terms is approximated by a linear finite element method with first-order accurate quadrature. The corresponding bilinear form does not need to be strongly positive. The discretisation is equivalent to a finite difference scheme. Although the discretisation is in general only first-order consistent, supraconvergence, i.e., convergence of higher order, is shown to take place even on nonuniform grids. If neither oblique boundary sections nor mixed derivatives occur, then the optimal order s is achieved. The supraconvergence result is equivalent to the supercloseness of the gradient.

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Global Superconvergence and A Posteriori Error Estimators of the Finite Element Method for a Quasi-linear Elliptic Boundary Value Problem of Nonmonotone Type

Cited by 11 publications

References 14 publications

A posteriori error estimates of edge residual type of finite element method for nonmonotone quasi‐linear elliptic problems

A posteriori error estimates of edge residual type of finite element method for nonmonotone quasi‐linear elliptic problems

Global Superconvergence Estimates of Petrov-Galerkin Methods for Hyperbolic Equations

Supraconvergence and Supercloseness of a Discretisation for Elliptic Third-kind Boundary-value Problems on Polygonal Domains

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