ABSTRACT
We describe how wavelets constructed out of finite-element interpolation functions provide a convenient mechanism for both error-estimation and adaptivity in finite-element analysis. This is done by posing an adaptive refinement problem as one of compactly representing a signal (the solution to the governing partial differential equation (PDE) or boundary integral equation (BIE), with isolated features of interest. To compress the solution in an efficient manner, we first compute approximately the details to be added to the solution on a coarse mesh in order to obtain the solution on a finer mesh (the estimation step) and then compute exactly the coefficients corresponding to only those basis functions contributing significantly to the details (the adaptation step). In this sense, therefore, the proposed approach is unified, since the basis functions used for error-estimation are exactly the same as those used for adaptive refinement. We illustrate the application of the proposed techniques for goal-oriented adaptivity for second- and fourth-order linear, elliptic PDEs.