A posteriori error analysis of a cell-centered finite volume method for semilinear elliptic problems

Estep, D.; Pernice, Michael; Pham, Du; Tavener, Simon; Wang, Haiying

doi:10.1016/j.cam.2009.07.046

Cited by 18 publications

(18 citation statements)

References 29 publications

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“…and (2) In each case, quantify the relative contributions of various aspects of discretization to the error in the computed information. The tool we use to address these issues is an adjoint-based a-posteriori error estimate Becker & Rannacher, 2001;Giles & Suli, 2002;Wheeler & Yotov, 2005;Estep et al, 2009a;Hansbro & Larson, 2011). This goal-oriented estimate accurately quantifies various contributions to the overall error.…”

Section: Discussionmentioning

confidence: 99%

“…To discretize, we use the lowest order Raviart-Thomas finite element space (RTO), in which the discrete scalar unknown p h is approximated as a constant over each cell, and the components of the discrete vector u h are approximated by functions that are piecewise linear in one spatial dimension and constant in the other (Bernardi et al, 2005;Estep et al, 2009a). The discrete interface unknown, £ ) h , is represented by piecewise discontinuous linears on the interface grid cells (Arbogast et al, 2000(Arbogast et al, , 2007.…”

Section: Of 24mentioning

confidence: 99%

“…It is important to distinguish residuals associated with approximating the solution spaces using finite dimensional polynomial spaces from residuals associated with approximating the integrals defining the variational formulation. We rewrite Galerkin orthogonality as Note that in the case of using the RTO finite element space and the midpoint-trapezoidal quadrature rules discussed above, the mixed finite element method reduces to the finite volume method (Russell & Wheeler, 1983;Weiser & Wheeler, 1988;Estep et al, 2009a), and some of the quadrature error terms are zero. These terms are included for generality, so that (3.10) is valid for other combinations of finite element spaces and quadratures.…”

Section: {R Ul U'^l) Q^{ R Pu P'^l) Q + (R Ull N^r) Q + (R Pit} P^kmentioning

confidence: 99%

See 2 more Smart Citations

A Posteriori Error Analysis and Uncertainty Quantification for Adaptive Multiscale Operator Decomposition Methods for Multiphysics Problems

Estep¹,

Holst²

2013

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Section: Discussionmentioning

confidence: 99%

Section: Of 24mentioning

confidence: 99%

Section: {R Ul U'^l) Q^{ R Pu P'^l) Q + (R Ull N^r) Q + (R Pit} P^kmentioning

confidence: 99%

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A Posteriori Error Analysis and Uncertainty Quantification for Adaptive Multiscale Operator Decomposition Methods for Multiphysics Problems

Estep¹,

Holst²

2013

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“…These adaptive procedures usually rely on a posteriori type error estimates of residuals [3][4][5][6] or quantities of interest [7]. Convergence of adaptive finite element methods for elliptic problems has been investigated for continuous finite elements in [3] and for discontinuous finite elements in [2,8].…”

Section: Introductionmentioning

confidence: 99%

An adaptive discontinuous finite volume method for elliptic problems

Liu¹,

Mu²,

Ye³

2011

Journal of Computational and Applied Mathematics

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MSC: 65N15 65N30 65N50Keywords: Adaptive mesh refinements a posteriori error estimates Elliptic boundary value problems Finite volume methods a b s t r a c t An adaptive discontinuous finite volume method is developed and analyzed in this paper. We prove that the adaptive procedure achieves guaranteed error reduction in a meshdependent energy norm and has a linear convergence rate. Numerical results are also presented to illustrate the theoretical analysis.

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“…The theory has been well developed using a variety of numerical methods for elliptic and parabolic problems, e.g. [10,18]. Numerous numerical studies exist for application to hyperbolic conservation laws, e.g.…”

Section: Introductionmentioning

confidence: 99%

Adjoint Error Estimation for Linear Advection

Connors¹,

Banks²,

Hittinger³

et al. 2011

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Abstract. An a posteriori error formula is described when a statistical measurement of the solution to a hyperbolic conservation law in 1D is estimated by finite volume approximations. This is accomplished using adjoint error estimation. In contrast to previously studied methods, the adjoint problem is divorced from the finite volume method used to approximate the forward solution variables. An exact error formula and computable error estimate are derived based on an abstractly defined approximation of the adjoint solution. This framework allows the error to be computed to an arbitrary accuracy given a sufficiently well resolved approximation of the adjoint solution. The accuracy of the computable error estimate provably satisfies an a priori error bound for sufficiently smooth solutions of the forward and adjoint problems. The theory does not currently account for discontinuities. Computational examples are provided that show support of the theory for smooth solutions. The application to problems with discontinuities is also investigated computationally.

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A posteriori error analysis of a cell-centered finite volume method for semilinear elliptic problems

Cited by 18 publications

References 29 publications

A Posteriori Error Analysis and Uncertainty Quantification for Adaptive Multiscale Operator Decomposition Methods for Multiphysics Problems

A Posteriori Error Analysis and Uncertainty Quantification for Adaptive Multiscale Operator Decomposition Methods for Multiphysics Problems

An adaptive discontinuous finite volume method for elliptic problems

Adjoint Error Estimation for Linear Advection

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