Adaptive Discontinuous Galerkin Methods with Multiwavelets Bases

We demonstrate the advantages of using multi-reolution analysis with multiwavelet basis with the Discontinuous Galerkin (DG) method. This provides significant enhancements to the standard DG methods. To illustrate the important gains of using the Multiwavelet DG method we apply it to conservation and convection diffusion problems in multiple dimensions. The significant benefits of merging DG methods with multiwavelets are three-fold. First, the DG method inherits a hierarchical structure from multiwavelets that produces a weak decoupling across different length scales. Second, hp-adaptivity in the DG method is naturally resolved through the multiwavelet basis rather than grid manipulation by the scaling properties of multiwavelets. Third, the matrix of the multiwavelet DG operator and its inverse share the same sparse pattern, that has the potential to provide nearly linear scaling of memory and computational performance with increasing degrees of freedom and dimensionality. In addition, the highly desired sparsity pattern combined with multiresolution provides a direct way for developing fast numerical solvers. These properties are especially important for higher dimensional problems with large degrees of freedom.