A Wavelet Based Adaptive Discontinuous Galerkin Method for Incompressible Flows

This paper outlines the development of a wavelet based adaptive discontinuous Galerkin spectral element method (DG-SEM) for unsteady incompressible flows. The proposed approach possesses arbitrary high formal accuracy and permits adaptivity in a way that is computationally cheap and efficient. An element wise discretisation of the domain is performed. Two sets of basis functions are employed per element-the Lagrange polynomials at the Gauss-Legendre-Lobatto (GLL) points which acts as the nodal basis for the DG-SEM method and the second generation wavelets (SGW) which can be looked upon as either a nodal or modal basis, subject to convenience, and is responsible for facilitating the adaptivity. The projection of the signal onto the wavelet space provides information about the local frequency content of the signal. An accumulation of high frequency components acts as an indicator for dynamic mesh refinement via thresholding. The main advantage of using the SGW basis is the low cost of the transform, O(N) per direction.