Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients

We present a posteriori error analysis of diffusion problems where the diffusion tensor is not necessarily symmetric and positive definite and can in particular change its sign. We first identify the correct intrinsic error norm for such problems, covering both conforming and nonconforming approximations. It combines a dual (residual) norm together with the distance to the correct functional space. Importantly, we show the equivalence of both these quantities defined globally over the entire computational domain with the Hilbertian sums of their localizations over patches of elements. In this framework, we then design a posteriori estimators which deliver simultaneously guaranteed error upper bound, global and local error lower bounds, and robustness with respect to the (sign-changing) diffusion tensor. Robustness with respect to the approximation polynomial degree is achieved as well. The estimators are given in a unified setting covering at once conforming, nonconforming, mixed, and discontinuous Galerkin finite element discretizations in two or three space dimensions. Numerical results illustrate the theoretical developments.

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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 ◽

10.1051/m2an/2020041 ◽

2020 ◽

Author(s):

Alexndre Ern ◽

Martin Vohralík ◽

Mohammad Zakerzadeh

Keyword(s):

Space Dimension ◽

A Posteriori Error Estimates ◽

Dual Graph ◽

Posteriori Error ◽

Posteriori Error Estimate ◽

Computational Domain ◽

A Posteriori ◽

Dual Norm ◽

A Posteriori Error ◽

Polynomial Degree

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 aligned and not aligned with the computational mesh.

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Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem

Mathematics of Computation ◽

10.1090/s0025-5718-2014-02854-8 ◽

2014 ◽

Vol 84 (291) ◽

pp. 153-186 ◽

Cited By ~ 10

Author(s):

Daniele A. Di Pietro ◽

Martin Vohralík ◽

Soleiman Yousef

Keyword(s):

Stefan Problem ◽

Error Control ◽

Posteriori Error ◽

A Posteriori ◽

Two Phase ◽

Adaptive Regularization ◽

A Posteriori Error

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Functional A Posteriori Error Control for Conforming Mixed Approximations of Coercive Problems with Lower Order Terms

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2016-0016 ◽

2016 ◽

Vol 16 (4) ◽

pp. 609-631 ◽

Cited By ~ 1

Author(s):

Immanuel Anjam ◽

Dirk Pauly

Keyword(s):

Error Control ◽

A Posteriori Error Estimates ◽

Reaction Diffusion ◽

Posteriori Error ◽

Diffusion Problem ◽

Mixed Formulation ◽

A Posteriori ◽

A Posteriori Error ◽

Linear Partial Differential Equations ◽

Primal And Dual

AbstractThe results of this contribution are derived in the framework of functional type a posteriori error estimates. The error is measured in a combined norm which takes into account both the primal and dual variables denoted by x and y, respectively. Our first main result is an error equality for all equations of the class ${\mathrm{A}^{*}\mathrm{A}x+x=f}$ or in mixed formulation ${\mathrm{A}^{*}y+x=f}$, ${\mathrm{A}x=y}$, where the exact solution $(x,y)$ is in $D(\mathrm{A})\times D(\mathrm{A}^{*})$. Here ${\mathrm{A}}$ is a linear, densely defined and closed (usually a differential) operator and ${\mathrm{A}^{*}}$ its adjoint. In this paper we deal with very conforming mixed approximations, i.e., we assume that the approximation ${(\tilde{x},\tilde{y})}$ belongs to ${D(\mathrm{A})\times D(\mathrm{A}^{*})}$. In order to obtain the exact global error value of this approximation one only needs the problem data and the mixed approximation itself, i.e., we have the equality$\lvert x-\tilde{x}\rvert^{2}+\lvert\mathrm{A}(x-\tilde{x})\rvert^{2}+\lvert y-% \tilde{y}\rvert^{2}+\lvert\mathrm{A}^{*}(y-\tilde{y})\rvert^{2}=\mathcal{M}(% \tilde{x},\tilde{y}),$where ${\mathcal{M}(\tilde{x},\tilde{y}):=\lvert f-\tilde{x}-\mathrm{A}^{*}\tilde{y}% \rvert^{2}+\lvert\tilde{y}-\mathrm{A}\tilde{x}\rvert^{2}}$ contains only known data. Our second main result is an error estimate for all equations of the class ${\mathrm{A}^{*}\mathrm{A}x+ix=f}$ or in mixed formulation ${\mathrm{A}^{*}y+ix=f}$, ${\mathrm{A}x=y}$, where i is the imaginary unit. For this problem we have the two-sided estimate$\frac{\sqrt{2}}{\sqrt{2}+1}\mathcal{M}_{i}(\tilde{x},\tilde{y})\leq\lvert x-% \tilde{x}\rvert^{2}+\lvert\mathrm{A}(x-\tilde{x})\rvert^{2}+\lvert y-\tilde{y}% \rvert^{2}+\lvert\mathrm{A}^{*}(y-\tilde{y})\rvert^{2}\leq\frac{\sqrt{2}}{% \sqrt{2}-1}\mathcal{M}_{i}(\tilde{x},\tilde{y}),$where ${\mathcal{M}_{i}(\tilde{x},\tilde{y}):=\lvert f-i\tilde{x}-\mathrm{A}^{*}% \tilde{y}\rvert^{2}+\lvert\tilde{y}-\mathrm{A}\tilde{x}\rvert^{2}}$ contains only known data. We will point out a motivation for the study of the latter problems by time discretizations or time-harmonic ansatz of linear partial differential equations and we will present an extensive list of applications including the reaction-diffusion problem and the eddy current problem.

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A Posteriori Error Control in Terms of Linear Functionals with Applications to Fracture Mechanics

International e-Conference of Computer Science 2006 ◽

10.1201/b12168-29 ◽

2007 ◽

pp. 135-138

Keyword(s):

Fracture Mechanics ◽

Error Control ◽

Posteriori Error ◽

A Posteriori ◽

Linear Functionals ◽

A Posteriori Error

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A priori and a posteriori error analysis for a hybrid formulation of a prestressed shell model.

10.22541/au.161639868.88765742/v1 ◽

2021 ◽

Author(s):

Serge Nicaise ◽

Ismail Merabet ◽

Rayhana REZZAG BARA

Keyword(s):

Shell Model ◽

A Priori ◽

Functional Space ◽

Finite Element Approximation ◽

Posteriori Error ◽

Error Estimator ◽

Element Approximation ◽

A Posteriori ◽

A Posteriori Error ◽

A Priori Error Estimate

This work deals with the finite element approximation of a prestressed shell model using a new formulation where the unknowns (the displacement and the rotation of fibers normal to the midsurface) are described in Cartesian and local covariant basis respectively. Due to the constraint involved in the definition of the functional space, a penalized version is then considered. We obtain a non robust a priori error estimate of this penalized formulation, but a robust one is obtained for its mixed formulation. Moreover, we present a reliable and efficient a posteriori error estimator of the penalized formulation. Numerical tests are included that confirmthe efficiency of our residual a posteriori estimator.

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