S. Repin

B. Khoromskij, S. Repin

We consider an iteration method for solving an elliptic type boundary value problem $\mathcal{A} u=f$, where a positive definite operator $\mathcal{A}$ is generated by a quasi--periodic structure with rapidly changing coefficients (typical period is characterized by a small parameter $ε$) . The method is based on using a simpler operator $\mathcal{A}_0$ (inversion of $\mathcal{A}_0$ is much simpler than inversion of $\mathcal{A}$), which can be viewed as a preconditioner for $\mathcal{A}$. We prove contraction of the iteration method and establish explicit estimates of the contraction factor $q$. Certainly the value of $q$ depends on the difference between $\mathcal{A}$ and $\mathcal{A}_0$. For typical quasi--periodic structures, we establish simple relations that suggest an optimal $\mathcal{A}_0$ (in a selected set of "simple" structures) and compute the corresponding contraction factor. Further, this allows us to deduce fully computable two--sided a posteriori estimates able to control numerical solutions on any iteration. The method is especially efficient if the coefficients of $\mathcal{A}$ admit low rank representations and algebraic operations are performed in tensor structured formats. Under moderate assumptions the storage and solution complexity of our approach depends only weakly (merely linear-logarithmically) on the frequency parameter $1/ε$, providing the FEM approximation of the order of $O(ε^{1+p})$, $p>0$.

M. Weymuth, S. Sauter, S. Repin

We consider elliptic problems with complicated, discontinuous diffusion tensor $A_{\scriptscriptstyle 0} $. One of the standard approaches to numerically treat such problems is to simplify the coefficient by some approximation, say $A_{\varepsilon}$, and to use standard finite elements. In \cite{Repin2012} a combined modelling-discretization strategy has been proposed which estimates the discretization and modelling errors by a posteriori estimates of functional type. This strategy allows to balance these two errors in a problem adapted way. However, the estimate of the modelling error is derived under the assumption that the difference $A_{\scriptscriptstyle 0} -A_{\varepsilon}$ is bounded in the $L^{\infty}$-norm, which requires that the approximation of the coefficient matches the discontinuities of the original coefficient. Therefore this theory is not appropriate for applications with discontinuous coefficients along \textit{complicated, curved} interfaces. Based on bounds for $A_{\scriptscriptstyle 0} -A_{\varepsilon}$ in an $L^{q}$-norm with $q<\infty$ we generalize the combined modelling-discretization strategy to a larger class of coefficients.

S. Repin

In the paper, we consider inequalities of the Poincaré--Steklov type for subspaces of $H^1$-functions defined in a bounded domain $Ω\in \Rd$ with Lipschitz boundary $\partialΩ$. For scalar valued functions, the subspaces are defined by zero mean condition on $\partialΩ$ or on a part of $\partialΩ$ having positive $d-1$ measure. For vector valued functions, zero mean conditions are imposed on components (e.g., normal components) of the function on certain $d-1$ dimensional manifolds (e.g., on plane or curvilinear faces of $\partialΩ$). We find explicit and simply computable bounds of the respective constants for domains typically used in finite element methods (triangles, quadrilaterals, tetrahedrons, prisms, pyramids, and domains composed of them). The second part of the paper discusses applications of the estimates to interpolation of scalar and vector valued functions. %383838

O. Mali, S. Repin

We consider linear reaction--diffusion problems with mixed Diriclet-Neumann-Robin conditions. The diffusion matrix, reaction coefficient, and the coefficient in the Robin boundary condition are defined with an uncertainty which allow bounded variations around some given mean values. A solution to such a problem cannot be exactly determined (it is a function in the set of "possible solutions" formed by generalized solutions related to possible data). The problem is to find parameters of this set. In this paper, we show that computable lower and upper bounds of the diameter (or radius) of the set can be expressed throughout problem data and parameters that regulate the indeterminacy range. Our method is based on using a posteriori error majorants and minorants of the functional type (see monographs Neittaanmäki&Repin 2004, Repin 2008), which explicitly depend on the coefficients and allow to obtain the corresponding lower and upper bounds by solving the respective extremal problems generated by indeterminacy of coefficients.