A posteriori error estimates in quantities of interest for the finite element heterogeneous multiscale method

Abstract

We present an a posteriori error analysis in quantities of interest for elliptic homogenization problems discretized by the finite element heterogeneous multiscale method. The multiscale method is based on a macro-to-micro formulation, where the macroscopic physical problem is discretized in a macroscopic finite element space and the missing macroscopic data is recovered on-the-fly using the solutions of corresponding microscopic problems. We propose a new framework that allows to follow the concept of the (single-scale) dual-weighted residual method at the macroscopic level in order to derive a posteriori error estimates in quantities of interests for multiscale problems. Local error indicators, derived in the macroscopic domain, can be used for adaptive goal-oriented mesh refinement. These error indicators rely only on available macroscopic and microscopic solutions. We further provide a detailed analysis of the data approximation error, including the quadrature errors. Numerical experiments confirm the efficiency of the adaptive method and the effectivity of our error estimates in the quantities of interest.

Adaptive reduced basis finite element heterogeneous multiscale method

“…In this section, we apply the RB technique to another multiscale adaptive method, the DWR FE-HMM [22], which is based on the framework of the dual-weighted residual method. Here we want to know the error in a certain quantity of interest, e.g., the value of the macro solution at a certain point, directional point-wise derivative of the macro solution or the average of the solution on a subdomain etc.…”

“…To guide the mesh refinement, we actually use the unsigned local refinement indicator defined asη H (K) = |η H (K)| (see [22]). …”

“…We sketch of the proof for this theorem that follows the steps of the proof of [22,Theorem 6]. For the first term of (4.56), we apply the decomposition,…”

“…A posteriori error analysis for the FE-HMM in the physical energy-norm has been derived in [21] and the extension of the FE-HMM to goal oriented adaptivity (in quantities of interest) has been discussed in [22]. For this latter adaptive method, the solution of dual problems (in a higher macro FE space) are required.…”

“…For this latter adaptive method, the solution of dual problems (in a higher macro FE space) are required. Both adaptive methods [21,22] have proved to be useful and more efficient than the FE-HMM with uniform refinement when the macroscopic domain or the macroscopic solution exhibit singularities. But as mentioned above, the simultaneous mesh refinement still represents a significant computational overhead that prevents to solve efficiently three-dimensional problems adaptively, or to use adaptive higher order macro FEM.…”