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Martin Vohralik - One of the best experts on this subject based on the ideXlab platform.
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Equivalence of local-and global-best approximations, a simple stable local commuting projector, and optimal hp approximation estimates in H(div)
IMA Journal of Numerical Analysis, 2021Co-Authors: Alexandre Ern, Thirupathi Gudi, Iain Smears, Martin VohralikAbstract:Given an arbitrary function in H(div), we show that the error attained by the global-best approximation by H(div)-conforming piecewise polynomial Raviart-Thomas-Nedelec elements under additional constraints on the divergence and normal flux on the boundary, is, up to a Generic Constant, equivalent to the sum of independent local-best approximation errors over individual mesh elements, without constraints on the divergence or normal fluxes. The Generic Constant only depends on the shape-regularity of the underlying simplicial mesh, the space dimension, and the polynomial degree of the approximations. The analysis also gives rise to a stable, local, commuting projector in H(div), delivering an approximation error that is equivalent to the local-best approximation. We next present a variant of the equivalence result, where robustness of the Constant with respect to the polynomial degree is attained for unbalanced approximations. These two results together further enable us to derive rates of convergence of global-best approximations that are fully optimal in both the mesh size h and the polynomial degree p, for vector fields that only feature elementwise the minimal necessary Sobolev regularity. We finally show how to apply our findings to derive optimal a priori hp-error estimates for mixed and least-squares finite element methods applied to a model diffusion problem.
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Equivalence of local-and global-best approximations, a simple stable local commuting projector, and optimal $hp$ approximation estimates in $H$(div)
IMA Journal of Numerical Analysis, 2021Co-Authors: Alexandre Ern, Thirupathi Gudi, Iain Smears, Martin VohralikAbstract:Given an arbitrary function in H(div), we show that the error attained by the global-best approximation by $H$(div)-conforming piecewise polynomial Raviart-Thomas-Nédélec elements under additional constraints on the divergence and normal flux on the boundary, is, up to a Generic Constant, equivalent to the sum of independent local-best approximation errors over individual mesh elements, without constraints on the divergence or normal fluxes. The Generic Constant only depends on the shape-regularity of the underlying simplicial mesh, the space dimension, and the polynomial degree of the approximations. The analysis also gives rise to a stable, local, commuting projector in $H$(div), delivering an approximation error that is equivalent to the local-best approximation. We next present a variant of the equivalence result, where robustness of the Constant with respect to the polynomial degree is attained for unbalanced approximations. These two results together further enable us to derive rates of convergence of global-best approximations that are fully optimal in both the mesh size $h$ and the polynomial degree $p$, for vector fields that only feature elementwise the minimal necessary Sobolev regularity. We finally show how to apply our findings to derive optimal a priori $hp$-error estimates for mixed and least-squares finite element methods applied to a model diffusion problem.
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Guaranteed and robust $L^2$-norm a posteriori error estimates for 1D linear advection problems
ESAIM: Mathematical Modelling and Numerical Analysis, 2021Co-Authors: Alexandre Ern, Martin Vohralik, Mohammad ZakerzadehAbstract:We propose a reconstruction-based a posteriori error estimate for linear advection problems in one space dimension. In our framework, a stable variational ultra-weak formulation is adopted, and the equivalence of the $L_2$-norm of the error with the dual graph norm of the residual is established. This dual norm is showed to be localizable over vertex-based patch subdomains of the computational domain under the condition of the orthogonality of the residual to the piecewise affine hat functions. We show that this condition is valid for some well-known numerical methods including continuous/discontinuous Petrov-Galerkin and discontinuous Galerkin methods. Consequently, a well-posed local problem on each patch is identified, which leads to a global conforming reconstruction of the discrete solution. We prove that this reconstruction provides a guaranteed upper bound on the $L_2$ error. Moreover, up to a Constant, it also gives local lower bounds on the $L_2$ error, where the Generic Constant is proven to be independent of mesh-refinement, polynomial degree of the approximation, and the advective velocity. This leads to robustness of our estimates with respect to the advection as well as the polynomial degree. All the above properties are verified in a series of numerical experiments, additionally leading to asymptotic exactness. Motivated by these results, we finally propose a heuristic extension of our methodology to any space dimension, achieved by solving local least-squares problems on vertex-based patches. Though not anymore guaranteed, the resulting error indicator is numerically robust with respect to both advection velocity and polynomial degree, for a collection of two-dimensional test cases including discontinuous solutions.
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Adaptive Inexact Semismooth Newton Methods for the Contact Problem Between Two Membranes
Journal of Scientific Computing, 2020Co-Authors: Jad Dabaghi, Vincent Martin, Martin VohralikAbstract:We propose an adaptive inexact version of a class of semismooth Newton methods that is aware of the continuous (variational) level. As a model problem, we study the system of variational inequalities describing the contact between two membranes. This problem is discretized with conforming finite elements of order $$p \ge 1$$ p ≥ 1 , yielding a nonlinear algebraic system with inequalities. We consider any iterative semismooth linearization algorithm like the Newton-min or the Newton–Fischer–Burmeister which we complement by any iterative linear algebraic solver. We then derive an a posteriori estimate on the error between the exact solution at the continuous level and the approximate solution which is valid at any step of the linearization and algebraic resolutions. Our estimate is based on flux reconstructions in discrete subspaces of $${\mathbf{H}}(\mathrm {div},\Omega )$$ H ( div , Ω ) and on potential reconstructions in discrete subspaces of $$H^1(\Omega )$$ H 1 ( Ω ) satisfying the constraints. It distinguishes the discretization, linearization, and algebraic components of the error. Consequently, we can formulate adaptive stopping criteria for both solvers, giving rise to an adaptive version of the considered inexact semismooth Newton algorithm. Under these criteria, the efficiency of the leading estimates is also established, meaning that we prove them equivalent with the error up to a Generic Constant. Numerical experiments for the Newton-min algorithm in combination with the GMRES algebraic solver confirm the efficiency of the developed adaptive method.
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Sharp algebraic and total a posteriori error bounds for h and p finite elements via a multilevel approach: Recovering mass balance in any situation
Computer Methods in Applied Mechanics and Engineering, 2020Co-Authors: Jan Papež, Martin Vohralik, Ulrich Rüde, Barbara WohlmuthAbstract:We present novel H(div) and H1 liftings of given piecewise polynomials over a hierarchy of simplicial meshes, based on a global solve on the coarsest mesh and on local solves on patches of mesh elements around vertices on subsequent mesh levels. This in particular allows to lift a given algebraic residual. In connection with approaches lifting the total residual, we show how to obtain guaranteed, fully computable, and Constant-free upper and lower a posteriori bounds on the algebraic, total, and discretization errors; here we consider the model Poisson equation discretized by the conforming finite element method of arbitrary order and including an arbitrary iterative solver. We next formulate safe stopping criteria ensuring that the algebraic error does not dominate the total error. We also prove efficiency, i.e., equivalence of our upper total and algebraic estimates with the total and algebraic errors, respectively, up to a Generic Constant; this Constant is polynomial-degree-independent for the total error. Numerical experiments illustrate sharp control of all error components and accurate prediction of their spatial distribution in several test problems, including cases where some classical estimators fail. The H(div)-liftings at the same time allow to recover mass balance for any problem, any numerical discretization, and any situation such as inexact solution of (nonlinear) algebraic systems or algorithm failure, which we believe is of independent interest. We demonstrate this mass balance recovery in a simulation of immiscible incompressible two-phase flow in porous media.
Ibrahim Cheddadi - One of the best experts on this subject based on the ideXlab platform.
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guaranteed and robust a posteriori error estimates for singularly perturbed reaction diffusion problems
Mathematical Modelling and Numerical Analysis, 2009Co-Authors: Ibrahim Cheddadi, Radek Fucik, Mariana I Prieto, Martin VohralikAbstract:We derive a posteriori error estimates for singularly perturbed reaction-diffusion problems which yield a guaranteed upper bound on the discretization error and are fully and easily computable. Moreover, they are also locally efficient and robust in the sense that they represent local lower bounds for the actual error, up to a Generic Constant independent in particular of the reaction coefficient. We present our results in the framework of the vertex-centered finite volume method but their nature is general for any conforming method, like the piecewise linear finite element one. Our estimates are based on a H(div)-conforming reconstruction of the diffusive flux in the lowest-order Raviart-Thomas space linked with mesh dual to the original simplicial one, previously introduced by the last author in the pure diffusion case. They also rely on elaborated Poincare, Friedrichs, and trace inequalities-based auxiliary estimates designed to cope optimally with the reaction dominance. In order to bring down the ratio of the estimated and actual overall energy error as close as possible to the optimal value of one, independently of the size of the reaction coefficient, we finally develop the ideas of local minimizations of the estimators by local modifications of the reconstructed diffusive flux. The numerical experiments presented confirm the guaranteed upper bound, robustness, and excellent efficiency of the derived estimates.
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Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems
ESAIM: Mathematical Modelling and Numerical Analysis, 2009Co-Authors: Ibrahim Cheddadi, Radek Fucik, Mariana Prieto, Martin VohralikAbstract:We derive a posteriori error estimates for singularly perturbed reaction-diffusion problems which yield a guaranteed upper bound on the discretization error and are fully and easily computable. Moreover, they are also locally efficient and robust in the sense that they represent local lower bounds for the actual error, up to a Generic Constant independent in particular of the reaction coefficient. We present our results in the framework of the vertex-centered finite volume method but their nature is general for any conforming method, like the piecewise linear finite element one. Our estimates are based on a H(div)-conforming reconstruction of the diffusive flux in the lowest-order Raviart-Thomas space linked with mesh dual to the original simplicial one, previously introduced by the last author in the pure diffusion case. They also rely on elaborated Poincaré, Friedrichs, and trace inequalities-based auxiliary estimates designed to cope optimally with the reaction dominance. In order to bring down the ratio of the estimated and actual overall energy error as close as possible to the optimal value of one, independently of the size of the reaction coefficient, we finally develop the ideas of local minimizations of the estimators by local modifications of the reconstructed diffusive flux. The numerical experiments presented confirm the guaranteed upper bound, robustness, and excellent efficiency of the derived estimates.
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Guaranteed and robust a posteriori error estimates for singularly perturbed reaction–diffusion problems
ESAIM: Mathematical Modelling and Numerical Analysis, 2009Co-Authors: Ibrahim Cheddadi, Radek Fucik, Mariana I Prieto, Martin VohralikAbstract:We derive a posteriori error estimates for singularly perturbed reaction-diffusion problems which yield a guaranteed upper bound on the discretization error and are fully and easily computable. Moreover, they are also locally efficient and robust in the sense that they represent local lower bounds for the actual error, up to a Generic Constant independent in particular of the reaction coefficient. We present our results in the framework of the vertex-centered finite volume method but their nature is general for any conforming method, like the piecewise linear finite element one. Our estimates are based on a H(div)-conforming reconstruction of the diffusive flux in the lowest-order Raviart- Thomas-Nedelec space linked with mesh dual to the original simplicial one, previously introduced by the last author in the pure diffusion case. They also rely on elaborated Poincare, Friedrichs, and trace inequalities-based auxiliary estimates designed to cope optimally with the reaction dominance. In order to bring down the ratio of the estimated and actual overall energy error as close as possible to the optimal value of one, independently of the size of the reaction coefficient, we finally develop the ideas of local minimizations of the estimators by local modifications of the reconstructed diffusive flux. The numerical experiments presented confirm the guaranteed upper bound, robustness, and excellent efficiency of the derived estimates.
Peter Wriggers - One of the best experts on this subject based on the ideXlab platform.
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Constant free explicit error estimator with sharp upper error bound property for adaptive fe analysis in elasticity and fracture
International Journal for Numerical Methods in Engineering, 2015Co-Authors: Tymofiy Gerasimov, E Stein, Peter WriggersAbstract:SUMMARY The explicit residual-based error estimator originally proposed by Babuska and Miller (1987) for adaptive finite element analysis with application to problems of linear elasticity and fracture is known to be one of the most simple and inexpensive error estimators. Indeed, it provides a theoretically guaranteed upper bound on a discretization error, measured in the energy norm, and requires small (nearly negligible) post-processing computational effort. The main issue with this classical estimator, however, is that an upper error bound is explicitly computable up to an unknown multiplicative Constant. In this paper, we track the source of this Generic Constant, revise the original derivation procedure and derive analytically four pre-computable Constants that constitute an upper error bound, resulting in, what we then call, the Constant-free error estimator of the Babuska-Miller type. The performance of this Constant-free estimator, as well as an adaptive FEM based on it, are illustrated on regular and singular benchmark problems and on numerical examples featuring crack propagation. The special property of the estimator is a sharp upper bound of the error: effectivity indices in the range of 1.2–2.0 are obtained, what is treated as practically (very) acceptable. In terms of efficiency judged against simplicity and inexpensiveness, the proposed Constant-free explicit estimator is superior to, for example, corresponding implicit residual estimators and may be seen as a very competitive one in comparison with other known, yet more intricate and complex error estimation techniques. Copyright © 2014 John Wiley & Sons, Ltd.
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Constant‐free explicit error estimator with sharp upper error bound property for adaptive FE analysis in elasticity and fracture
International Journal for Numerical Methods in Engineering, 2014Co-Authors: Tymofiy Gerasimov, E Stein, Peter WriggersAbstract:SUMMARY The explicit residual-based error estimator originally proposed by Babuska and Miller (1987) for adaptive finite element analysis with application to problems of linear elasticity and fracture is known to be one of the most simple and inexpensive error estimators. Indeed, it provides a theoretically guaranteed upper bound on a discretization error, measured in the energy norm, and requires small (nearly negligible) post-processing computational effort. The main issue with this classical estimator, however, is that an upper error bound is explicitly computable up to an unknown multiplicative Constant. In this paper, we track the source of this Generic Constant, revise the original derivation procedure and derive analytically four pre-computable Constants that constitute an upper error bound, resulting in, what we then call, the Constant-free error estimator of the Babuska-Miller type. The performance of this Constant-free estimator, as well as an adaptive FEM based on it, are illustrated on regular and singular benchmark problems and on numerical examples featuring crack propagation. The special property of the estimator is a sharp upper bound of the error: effectivity indices in the range of 1.2–2.0 are obtained, what is treated as practically (very) acceptable. In terms of efficiency judged against simplicity and inexpensiveness, the proposed Constant-free explicit estimator is superior to, for example, corresponding implicit residual estimators and may be seen as a very competitive one in comparison with other known, yet more intricate and complex error estimation techniques. Copyright © 2014 John Wiley & Sons, Ltd.
Alexandre Ern - One of the best experts on this subject based on the ideXlab platform.
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Quasi-optimal nonconforming approximation of elliptic PDES with contrasted coefficients and $H^{1+r}$, $r>0$, regularity
2021Co-Authors: Alexandre Ern, Jean-luc GuermondAbstract:In this paper, we investigate the approximation of a diffusion model problem with contrasted diffusivity for various nonconforming approximation methods. The essential difficulty is that the Sobolev smoothness index of the exact solution may be just barely larger than 1. The lack of smoothness is handled by giving a weak meaning to the normal derivative of the exact solution at the mesh faces. We derive robust and quasi-optimal error estimates. Quasi-optimality means that the approximation error is bounded, up to a Generic Constant, by the best-approximation error in the discrete trial space, and robustness means that the Generic Constant is independent of the diffusivity contrast. The error estimates use a mesh-dependent norm that is equivalent, at the discrete level, to the energy norm and that remains bounded as long as the exact solution has a Sobolev index strictly larger than 1. Finally, we briefly show how the analysis can be extended to the Maxwell's equations.
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Equivalence of local-and global-best approximations, a simple stable local commuting projector, and optimal hp approximation estimates in H(div)
IMA Journal of Numerical Analysis, 2021Co-Authors: Alexandre Ern, Thirupathi Gudi, Iain Smears, Martin VohralikAbstract:Given an arbitrary function in H(div), we show that the error attained by the global-best approximation by H(div)-conforming piecewise polynomial Raviart-Thomas-Nedelec elements under additional constraints on the divergence and normal flux on the boundary, is, up to a Generic Constant, equivalent to the sum of independent local-best approximation errors over individual mesh elements, without constraints on the divergence or normal fluxes. The Generic Constant only depends on the shape-regularity of the underlying simplicial mesh, the space dimension, and the polynomial degree of the approximations. The analysis also gives rise to a stable, local, commuting projector in H(div), delivering an approximation error that is equivalent to the local-best approximation. We next present a variant of the equivalence result, where robustness of the Constant with respect to the polynomial degree is attained for unbalanced approximations. These two results together further enable us to derive rates of convergence of global-best approximations that are fully optimal in both the mesh size h and the polynomial degree p, for vector fields that only feature elementwise the minimal necessary Sobolev regularity. We finally show how to apply our findings to derive optimal a priori hp-error estimates for mixed and least-squares finite element methods applied to a model diffusion problem.
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Equivalence of local-and global-best approximations, a simple stable local commuting projector, and optimal $hp$ approximation estimates in $H$(div)
IMA Journal of Numerical Analysis, 2021Co-Authors: Alexandre Ern, Thirupathi Gudi, Iain Smears, Martin VohralikAbstract:Given an arbitrary function in H(div), we show that the error attained by the global-best approximation by $H$(div)-conforming piecewise polynomial Raviart-Thomas-Nédélec elements under additional constraints on the divergence and normal flux on the boundary, is, up to a Generic Constant, equivalent to the sum of independent local-best approximation errors over individual mesh elements, without constraints on the divergence or normal fluxes. The Generic Constant only depends on the shape-regularity of the underlying simplicial mesh, the space dimension, and the polynomial degree of the approximations. The analysis also gives rise to a stable, local, commuting projector in $H$(div), delivering an approximation error that is equivalent to the local-best approximation. We next present a variant of the equivalence result, where robustness of the Constant with respect to the polynomial degree is attained for unbalanced approximations. These two results together further enable us to derive rates of convergence of global-best approximations that are fully optimal in both the mesh size $h$ and the polynomial degree $p$, for vector fields that only feature elementwise the minimal necessary Sobolev regularity. We finally show how to apply our findings to derive optimal a priori $hp$-error estimates for mixed and least-squares finite element methods applied to a model diffusion problem.
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Guaranteed and robust $L^2$-norm a posteriori error estimates for 1D linear advection problems
ESAIM: Mathematical Modelling and Numerical Analysis, 2021Co-Authors: Alexandre Ern, Martin Vohralik, Mohammad ZakerzadehAbstract:We propose a reconstruction-based a posteriori error estimate for linear advection problems in one space dimension. In our framework, a stable variational ultra-weak formulation is adopted, and the equivalence of the $L_2$-norm of the error with the dual graph norm of the residual is established. This dual norm is showed to be localizable over vertex-based patch subdomains of the computational domain under the condition of the orthogonality of the residual to the piecewise affine hat functions. We show that this condition is valid for some well-known numerical methods including continuous/discontinuous Petrov-Galerkin and discontinuous Galerkin methods. Consequently, a well-posed local problem on each patch is identified, which leads to a global conforming reconstruction of the discrete solution. We prove that this reconstruction provides a guaranteed upper bound on the $L_2$ error. Moreover, up to a Constant, it also gives local lower bounds on the $L_2$ error, where the Generic Constant is proven to be independent of mesh-refinement, polynomial degree of the approximation, and the advective velocity. This leads to robustness of our estimates with respect to the advection as well as the polynomial degree. All the above properties are verified in a series of numerical experiments, additionally leading to asymptotic exactness. Motivated by these results, we finally propose a heuristic extension of our methodology to any space dimension, achieved by solving local least-squares problems on vertex-based patches. Though not anymore guaranteed, the resulting error indicator is numerically robust with respect to both advection velocity and polynomial degree, for a collection of two-dimensional test cases including discontinuous solutions.
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hp adaptation driven by polynomial degree robust a posteriori error estimates for elliptic problems
SIAM Journal on Scientific Computing, 2016Co-Authors: Vit Dolejsi, Alexandre Ern, Martin VohralikAbstract:We devise and study experimentally adaptive strategies driven by a posteriori error estimates to select automatically both the space mesh and the polynomial degree in the numerical approximation of diffusion equations in two space dimensions. The adaptation is based on equilibrated flux estimates. These estimates are presented here for inhomogeneous Dirichlet and Neumann boundary conditions, for spatially-varying polynomial degree, and for mixed rectangular-triangular grids possibly containing hanging nodes. They deliver a global error upper bound with Constant one and, up to data oscillation, error lower bounds on element patches with a Generic Constant only dependent on the mesh regularity and with a computable bound. We numerically asses the estimates and several hp-adaptive strategies using the interior penalty discontinuous Galerkin method. Asymptotic exactness is observed for all the symmetric, nonsymmetric (odd degrees), and incomplete variants on non-nested unstructured triangular grids for a smooth solution and uniform refinement. Exponential convergence rates are reported on nonmatching triangular grids for the incomplete version on several benchmarks with a singular solution and adaptive refinement.
Radek Fucik - One of the best experts on this subject based on the ideXlab platform.
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guaranteed and robust a posteriori error estimates for singularly perturbed reaction diffusion problems
Mathematical Modelling and Numerical Analysis, 2009Co-Authors: Ibrahim Cheddadi, Radek Fucik, Mariana I Prieto, Martin VohralikAbstract:We derive a posteriori error estimates for singularly perturbed reaction-diffusion problems which yield a guaranteed upper bound on the discretization error and are fully and easily computable. Moreover, they are also locally efficient and robust in the sense that they represent local lower bounds for the actual error, up to a Generic Constant independent in particular of the reaction coefficient. We present our results in the framework of the vertex-centered finite volume method but their nature is general for any conforming method, like the piecewise linear finite element one. Our estimates are based on a H(div)-conforming reconstruction of the diffusive flux in the lowest-order Raviart-Thomas space linked with mesh dual to the original simplicial one, previously introduced by the last author in the pure diffusion case. They also rely on elaborated Poincare, Friedrichs, and trace inequalities-based auxiliary estimates designed to cope optimally with the reaction dominance. In order to bring down the ratio of the estimated and actual overall energy error as close as possible to the optimal value of one, independently of the size of the reaction coefficient, we finally develop the ideas of local minimizations of the estimators by local modifications of the reconstructed diffusive flux. The numerical experiments presented confirm the guaranteed upper bound, robustness, and excellent efficiency of the derived estimates.
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Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems
ESAIM: Mathematical Modelling and Numerical Analysis, 2009Co-Authors: Ibrahim Cheddadi, Radek Fucik, Mariana Prieto, Martin VohralikAbstract:We derive a posteriori error estimates for singularly perturbed reaction-diffusion problems which yield a guaranteed upper bound on the discretization error and are fully and easily computable. Moreover, they are also locally efficient and robust in the sense that they represent local lower bounds for the actual error, up to a Generic Constant independent in particular of the reaction coefficient. We present our results in the framework of the vertex-centered finite volume method but their nature is general for any conforming method, like the piecewise linear finite element one. Our estimates are based on a H(div)-conforming reconstruction of the diffusive flux in the lowest-order Raviart-Thomas space linked with mesh dual to the original simplicial one, previously introduced by the last author in the pure diffusion case. They also rely on elaborated Poincaré, Friedrichs, and trace inequalities-based auxiliary estimates designed to cope optimally with the reaction dominance. In order to bring down the ratio of the estimated and actual overall energy error as close as possible to the optimal value of one, independently of the size of the reaction coefficient, we finally develop the ideas of local minimizations of the estimators by local modifications of the reconstructed diffusive flux. The numerical experiments presented confirm the guaranteed upper bound, robustness, and excellent efficiency of the derived estimates.
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Guaranteed and robust a posteriori error estimates for singularly perturbed reaction–diffusion problems
ESAIM: Mathematical Modelling and Numerical Analysis, 2009Co-Authors: Ibrahim Cheddadi, Radek Fucik, Mariana I Prieto, Martin VohralikAbstract:We derive a posteriori error estimates for singularly perturbed reaction-diffusion problems which yield a guaranteed upper bound on the discretization error and are fully and easily computable. Moreover, they are also locally efficient and robust in the sense that they represent local lower bounds for the actual error, up to a Generic Constant independent in particular of the reaction coefficient. We present our results in the framework of the vertex-centered finite volume method but their nature is general for any conforming method, like the piecewise linear finite element one. Our estimates are based on a H(div)-conforming reconstruction of the diffusive flux in the lowest-order Raviart- Thomas-Nedelec space linked with mesh dual to the original simplicial one, previously introduced by the last author in the pure diffusion case. They also rely on elaborated Poincare, Friedrichs, and trace inequalities-based auxiliary estimates designed to cope optimally with the reaction dominance. In order to bring down the ratio of the estimated and actual overall energy error as close as possible to the optimal value of one, independently of the size of the reaction coefficient, we finally develop the ideas of local minimizations of the estimators by local modifications of the reconstructed diffusive flux. The numerical experiments presented confirm the guaranteed upper bound, robustness, and excellent efficiency of the derived estimates.