An Adaptive Wavelet Method for Semi-Linear First-Order System Least Squares

We design an adaptive wavelet scheme for solving first-order system least-squares formulations of second-order elliptic PDEs that converge with the best possible rate in linear complexity. A wavelet Riesz basis is constructed for the space $\vec{H}_{0,\Gamma _N}(\operatorname{div};\Omega )$ on general polygons. The theoretical findings are illustrated by numerical experiments.

A Dynamic Adaptive Wavelet Method for Solution of the Schrodinger Equation

In this paper, we present a dynamically adaptive wavelet method for solving Schrodinger equation on one-dimensional, two-dimensional and on the sphere. Solving one-dimensional and two-dimensional Schrodinger equations are based on Daubechies wavelet with finite difference method on an arbitrary grid, and for spherical Schrodinger equation is based on spherical wavelet over an optimal spherical geodesic grid. The method is applied to the solution of Schrodinger equation for computational efficiency and achieve accuracy with controlling spatial grid adaptation — high resolution computations are performed only in regions where a solution varies greatly (i.e., near steep gradients, or near-singularities) and a much coarser grid where the solution varies slowly. Thereupon the dynamic adaptive wavelet method is useful to analyze local structure of solution with very less number of computational cost than any other methods. The prowess and computational efficiency of the adaptive wavelet method is demonstrated for the solution of Schrodinger equation on one-dimensional, two-dimensional and on the sphere.