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, where typically the coefficients are smooth or piecewise constant. 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. For this reason, inversion of the original operator associated with oscillating coefficients is avoided. The only operation required instead is multiplication of it on a vector (vector function), which can be efficiently performed due to the low QTT-rank tensor operations with the rank parameter controlled by the given precision δ > 0 independent on the parameter ε. We prove that solutions generated by the iteration method converge to the solution of the original problem provided that the parameter of the iteration algorithm has been properly selected. Moreover, we deduce two-sided a posteriori error estimates that do not use A_ε⁻¹ and enable us to compute guaranteed bounds of the distance to the exact solution of the original problem for any step of the iteration process. 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 the frequency parameter 1/ε. Numerical tests in 1D confirm the logarithmic complexity O(|log_ε|) of the proposed method applied to a class of complicated highly-oscillating coefficients.