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MiS Preprint
10/2017
Rank structured approximation method for quasi--periodic elliptic problems
Boris N. Khoromskij and Sergey Repin
Abstract
We consider an iteration method for solving an elliptic type boundary value problem $Au=f$, where a positive definite operator $A$ is generated by a quasi--periodic structure with rapidly changing coefficients (typical period is characterized by a small parameter $\epsilon$) . The method is based on using a simpler operator $A_0$ (inversion of $A_0$ is much simpler than inversion of $A$), which can be viewed as a preconditioner for $A$. 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 $A$ and $A_0$. For typical quasi--periodic structures, we establish simple relations that suggest an optimal $A_0$ (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 $A$ admit low rank representations and algebraic operations are performed by tensor type methods. Under moderate assumptions the storage and solution complexity of our approach depends only weakly (merely logarithmically) on the frequency parameter $1/\epsilon$, providing the FEM approximation of the order of $O(\epsilon^{1+q})$, $q>0$.
