Adaptive Discontinuous Galerkin Methods with Multiwavelets Bases

نویسندگان

  • Rick Archibald
  • George Fann
  • William Shelton
چکیده

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.

برای دانلود رایگان متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Adaptive Multiresolution Discontinuous Galerkin Schemes for Conservation Laws: Multi–Dimensional Case

The concept of multiresolution-based adaptive DG schemes for nonlinear one-dimensional hyperbolic conservation laws has been developed and investigated analytically and numerically in N. Hovhannisyan, S. Müller, R. Schäfer, Adaptive multiresolution Discontinuous Galerkin Schemes for Conservation Laws, Math. Comp., 2013. The key idea is to perform a multiresolution analysis using multiwavelets o...

متن کامل

Convergence Analysis of the Lowest Order Weakly Penalized Adaptive Discontinuous Galerkin Methods

In this article, we prove convergence of the weakly penalized adaptive discontinuous Galerkin methods. Unlike other works, we derive the contraction property for various discontinuous Galerkin methods only assuming the stabilizing parameters are large enough to stabilize the method. A central idea in the analysis is to construct an auxiliary solution from the discontinuous Galerkin solution by ...

متن کامل

Time Acceleration Methods for Advection on the Cubed Sphere

Climate simulation will not grow to the ultrascale without new algorithms to overcome the scalability barriers blocking existing implementations. Until recently, climate simulations concentrated on the question of whether the climate is changing. The emphasis is now shifting to impact assessments, mitigation and adaptation strategies, and regional details. Such studies will require significant ...

متن کامل

A High-Order Discontinuous Galerkin Discretization with Multiwavelet-Based Grid Adaptation for Compressible Flows

The modern vision of a flow solver necessarily includes adaptivity. In particular, mesh adaptivity enables the solution strategy to allocate the resources efficiently, in that cells are concentrated in areas where they are needed, as opposed to uniform mesh refinement. Multiresolution-based mesh adaptivity using biorthogonal wavelets has been quite successful with finite volume solvers for comp...

متن کامل

Multiwavelets for Second-kind Integral Equations

Abstract. We consider a Galerkin method for an elliptic pseudodifferential operator of order zero on a two-dimensional manifold. We use piecewise linear discontinuous trial functions on a triangular mesh and describe an orthonormal wavelet basis. Using this basis we can compress the stiffness matrix from N to O(N logN) nonzero entries and still obtain (up to logN terms) the same convergence rat...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2009