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

Abstract

The 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.

Functional a posteriori error estimates for boundary element methods

“…The identities (6) generate a posteriori error estimates of the functional type that are independent of the discretization and provide fully guaranteed lower and upper bounds for the unknown error without any constants at all. In general, these functional type a posteriori estimates involve only constants in basic functional inequalities associated with the concrete problem (e.g., Poincaré-Friedrichs type or trace inequalities) and are applicable for any approximation from the admissible energy class (see [1,2,37,39] or the monograph [40] and the references cited therein). In particular, the equations (6) have also been used in [40] for the analysis of errors arising in the Trefftz method.…”

An Elementary Method of Deriving A Posteriori Error Equalities and Estimates for Linear Partial Differential Equations

“…In this paper we expose a way of deriving functional type error equalities and estimates which does not require knowledge of variational or weak formulations of the corresponding problems, and involves only elementary operations. This method was already used in [1] for static problems with lower order terms, and is extended in the present work to problems without lower order terms, and, more interestingly, to parabolic problems. We note that the method presented here will produce a posteriori error equalities and estimates for mixed approximations, i.e., the error will be measured in a combined norm taking into account both the primal and the dual variable.…”

Fast MATLAB assembly of FEM matrices in 2D and 3D: Edge elements

“…Elliptic problems containing the full gradient operator ∇ of scalar or vector arguments are formulated in weak forms in H 1 Sobolev spaces and discretized using nodal finite element functions. Efficient MATLAB vectorization of the assembly routine of stiffness matrices for the linear nodal finite element was explained by T. Rahman and J. Valdman in [11].…”