Multimesh and multiresolution analysis for mesh adaptive interpolation Original Research Article

Let us call a highly heterogeneous function a function that is either locally singular or a smooth function but, with very small details in comparison with domain size. We first study the L2-norm of the interpolation error Eh between a function u and Πhu, its P1-continuous interpolate: we use four examples of functions that represent different cases of highly heterogeneous functions. When a sequence of uniform meshes is chosen, if we examine the convergence of Eh as a function of number of nodes, we observe a convergence of order 2 only for a smooth function and when the number of nodes is large enough. Conversely, when an adaptive mesh sequence is applied, second-order convergence is almost always observed. We give some theoretical arguments concerning this phenomenon. Secondly, following some ideas currently used in spectral and wavelet methods, we consider the P1-approximation of u on nested meshes and express the representation of uh as a series with increasing fineness of its terms. The size of each terms as a function of the corresponding level number is examined in relation with mesh adaption.