Adaptive multiwavelet-hierarchical method for multiscale computation

An adaptive multiwavelet-hierarchical method characterized by high convergent rate and flexible adaptive strategy is proposed for multiscale computation of field problems. According to the Strang - Fix condition, the convergence rate of the finite element multiwavelet method is determined by the approximation order of scaling functions in the same level of multiwavelet refinement. To raise the approximation order of scaling functions, finite element multiwavelets are combined with hierarchical bases to construct a new multilevel multiwavelet-hierarchical space. An adaptive strategy for multiwavelet-hierarchical refinement is presented based on new error estimation in the form of multiwavelets and hierarchical bases, which leaves much freedom for the problem-oriented selection of multiwavelets or hierarchical functions. Numerical examples demonstrate that the proposed method is an accurate and efficient tool in solving the field problems with singularities or changes in high gradients.