A fast iteration method for solving elliptic problems with quasiperiodic coefficients

Abstract

The paper suggests a preconditioning type method for fast solving of elliptic equations with oscillating quasiperiodic coefficients $A_ε$ specified by the small parameter $ε>0$. We use an iteration method generated by an elliptic operator, associated with a certain simplified (e.g., homogenized) problem. On each step of this procedure it is required to solve an auxiliary elliptic boundary value problem with non--oscillating coefficients $A_0$. All the information related to complicated coefficients of the original differential problem is encompasses in the linear functional, which forms the right hand side of the auxiliary problem. Therefore, explicit inversion of the original operator associated with oscillating coefficients is avoided. The only operation used instead is multiplication of the operator by a vector (vector function), which can be efficiently performed due to the low-rank QTT tensor operations with the rank parameter controlled by the given precision $δ>0$ independent on the parameter $ε$. We deduce two--sided a posteriori error estimates that do not use $A^{-1}_ε$ and provide guaranteed two sided bounds of the distance to the exact solution of the original problem for any step of the iteration process. The second part is concerned with realisations of the iteration method. For a wide class of oscillating coefficients, we obtain sharp QTT rank estimates for the stiffness matrix in tensor representation. In practice, this leads to the logarithmic complexity scaling of the approximation and solution process in both the FEM grid-size, and $O(\vert\logε\vert)$ cost in terms of $ε$. Numerical tests in 1D confirm the logarithmic complexity scaling of our method applied to a class of complicated quasiperiodic coefficients.