Dissipation in adaptive wavelet Galerkin discretizations

Adaptive wavelet schemes for solving partial differential equations offer an attractive possibility to introduce locally refined grids, which dynamically track the evolution of the solution in scale and space. Automatic error control of the adaptive discretization, with respect to a uniform grid solution, is an advantageous feature. Here we focus on dynamical Galerkin schemes, where the projection operator changes over time. When selecting a subset of basis functions, the projection operator is non-differentiable and an integral formulation has to be used. We will analyze the projected equations with respect to existence and uniqueness of the solution and prove that non-smooth projection operators introduce dissipation, a result which is crucial for adaptive discretizations of PDEs. For the Burgers equation we will illustrate numerically that thresholding the wavelet coefficients, and thus changing the projection space, will indeed introduce dissipation of energy. We discuss consequences for adaptive simulations of the incompressible Euler equations in two and three space dimensions.