Adaptive discontinuous Galerkin methods in multiwavelets bases (original) (raw)
Related papers
The numerical performance of wavelets for PDEs: the multi-scale finite element
Computational Mechanics, 2000
The research summarized in this paper is part of a multi-year effort focused on evaluating the viability of wavelet bases for the solution of partial differential equations. The primary objective for this work has been to establish a foundation for hierarchical/wavelet simulation methods based upon numerical performance, computational ef®ciency, and the ability to exploit the hierarchical adaptive nature of wavelets. This work has demonstrated that hierarchical bases can be effective for problems with a dominant elliptic character. However, the strict enforcement of orthogonality in the usual L 2 sense is less desirable than orthogonality in the energy norm. This conclusion has led to the development of a multi-scale linear ®nite element based on a hierarchical change-of-basis. This work considers the numerical and computational performance of the hierarchical Schauder basis in a Galerkin context. A unique row-column lumping procedure is developed with multi-scale solution strategies for 1-D and 2-D elliptic partial differential equations.
Computational and Applied Mathematics, 2014
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 on a hierarchy of nested grids for the data given on a uniformly refined mesh. This provides difference information between successive refinement levels that may become negligibly small in regions where the solution is locally smooth. Applying hard thresholding the data are highly compressed and local grid adaptation is triggered by the remaining significant coefficients. The focus of the present work lies on the extension of the originally one-dimensional concept to higher dimensions and the verification of the choice for the threshold value by means of parameter studies performed for linear and nonlinear scalar conservation laws.
A Wavelet Optimized Adaptive Multi-domain Method
Journal of Computational Physics, 1998
The formulation and implementation of wavelet based methods for the solution of multidimensional partial differential equations in complex geometries is discussed. Utilizing the close connection between Daubechies wavelets and finite difference methods on arbitrary grids, we formulate a wavelet based collocation method, well suited for dealing with general boundary conditions and nonlinearities.
Advances in Computational Mathematics, 2020
This paper is concerned with the construction, analysis and realization of a numerical method to approximate the solution of high dimensional elliptic partial differential equations. We propose a new combination of an Adaptive Wavelet Galerkin Method (AWGM) and the wellknown Hierarchical Tensor (HT) format. The arising HT-AWGM is adaptive both in the wavelet representation of the low dimensional factors and in the tensor rank of the HT representation. The point of departure is an adaptive wavelet method for the HT format using approximate Richardson iterations from [1] and an AWGM method as described in [13]. HT-AWGM performs a sequence of Galerkin solves based upon a truncated preconditioned conjugate gradient (PCG) algorithm from [33] in combination with a tensor-based preconditioner from [3]. Our analysis starts by showing convergence of the truncated conjugate gradient method. The next step is to add routines realizing the adaptive refinement. The resulting HT-AWGM is analyzed concerning convergence and complexity. We show that the performance of the scheme asymptotically depends only on the desired tolerance with convergence rates depending on the Besov regularity of low dimensional quantities and the low rank tensor structure of the solution. The complexity in the ranks is algebraic with powers of four stemming from the complexity of the tensor truncation. Numerical experiments show the quantitative performance.
Wavelets and adaptive grids for the discontinuous Galerkin method
Numerical Algorithms, 2005
In this paper, space adaptivity is introduced to control the error in the numerical solution of hyperbolic systems of conservation laws. The reference numerical scheme is a new version of the discontinuous Galerkin method, which uses an implicit diffusive term in the direction of the streamlines, for stability purposes. The decision whether to refine or to unrefine the grid in a certain location is taken according to the magnitude of wavelet coefficients, which are indicators of local smoothness of the numerical solution. Numerical solutions of the nonlinear Euler equations illustrate the efficiency of the method.
A discontinuous Galerkin multiscale method for convection-diffusion problems
We propose an extension of the discontinuous Galerkin local orthogonal decompo-sition multiscale method, presented in [14], to convection-diffusion problems with rough, heteroge-neous, and highly varying coefficients. The properties of the multiscale method and the discontinuous Galerkin method allows us to better cope with multiscale features as well as interior/boundary lay-ers in the solution. In the proposed method the trail and test spaces are spanned by a corrected basis computed on localized patches of size O(H log(H −1)), where H is the mesh size. We prove convergence rates independent of the variation in the coefficients and present numerical experiments which verify the analytic findings.
A Hybrid Scheme Based Upon Non-Dyadic Wavelets for the Solution of linear Sobolev Equations
Journal of emerging technologies and innovative research, 2019
The current work aims to introduce a hybrid technique based upon the θ −weighted differencing and non-dyadic wavelets for the solution of (2+1)-dimensional linear Sobolev Equations etc. In the scheme, time discretization is done by the θ −weighted finite differencing scheme and spatial discretization is done by Non-dyadic wavelets. The proposed scheme is tested on five different linear and nonlinear equations of above said types to establish the competency of the proposed scheme.
Eprint Arxiv Physics 0209040, 2002
It is shown how various ideas that are well established for the solution of Poisson's equation using plane wave and multigrid methods can be combined with wavelet concepts. The combination of wavelet concepts and multigrid techniques turns out to be particularly fruitful. We propose a modified multigrid V cycle scheme that is not only much simpler, but also more efficient than the standard V cycle. Whereas in the traditional V cycle the residue is passed to the coarser grid levels, this new scheme does not require the calculation of a residue. Instead it works with copies of the charge density on the different grid levels that were obtained from the underlying charge density on the finest grid by wavelet transformations. This scheme is not limited to the pure wavelet setting, where it is faster than the preconditioned conjugate gradient method, but equally well applicable for finite difference discretizations.