An Efficient Approximate Residual Evaluation in the Adaptive Tensor Product Wavelet Method

A wide class of well-posed operator equations can be solved in optimal computational complexity by adaptive wavelet methods. A quantitative bottleneck is the approximate evaluation of the arising residuals that steer the adaptive refinements. In this paper, we consider multi-tree approximations from tensor product wavelet bases for solving linear PDE’s. In this setting, we develop a new efficient approximate residual evaluation. Other than the commonly applied method, that uses the so-called APPLY routine, our approximate residual depends affinely on the current approximation of the solution. Our findings are illustrated by numerical results that show a considerable speed-up.