Error Analysis

Abstract

The objective of the proposed adaptive scheme is to reduce for a given FVS computational cost and memory requirements while preserving the accuracy of the reference scheme. In order to quantify this we introduce the averages $\hat u_L^n$ of the exact solution, the averages v ⁿ _L determined by the FVS and the averages $\bar v_L^n$ of the adaptive scheme. Note, that all sequences correspond to the finest discretization level indicated by L. In particular, the sequence $\bar v_L^n$ is not generated by the adaptive scheme on the full uniform highest level L but these data are determined by the adaptive array $\left\{ {v_{j,k}^n } \right\}_{(j,k) \in \mathcal{G}_{L,\varepsilon }^n }$ and could be retrieved from them by means of the local inverse multiscale transformation (3.21) — (3.23) where the non-significant details d _j,k,e in (3.23) are put to zero.