Capitalizing on Superconvergence for More Accurate Multi-Resolution Discontinuous Galerkin Methods

Galerkin Methods ◽

Accurate Approximation ◽

Siac Filtering ◽

Multi Resolution Analysis ◽

Resolution Analysis

AbstractThis article focuses on exploiting superconvergence to obtain more accurate multi-resolution analysis. Specifically, we concentrate on enhancing the quality of passing of information between scales by implementing the Smoothness-Increasing Accuracy-Conserving (SIAC) filtering combined with multi-wavelets. This allows for a more accurate approximation when passing information between meshes of different resolutions. Although this article presents the details of the SIAC filter using the standard discontinuous Galerkin method, these techniques are easily extendable to other types of data.

A Modified Discontinuous Galerkin Method for Solving Efficiently Helmholtz Problems

Communications in Computational Physics ◽

10.4208/cicp.081209.070710s ◽

2012 ◽

Vol 11 (2) ◽

pp. 335-350 ◽

Cited By ~ 4

Author(s):

Magdalena Grigoroscuta-Strugaru ◽

Mohamed Amara ◽

Henri Calandra ◽

Rabia Djellouli

Keyword(s):

Galerkin Method ◽

Discontinuous Galerkin ◽

Galerkin Methods ◽

Two Dimensional ◽

Local Problems ◽

The Stability ◽

Well Posed ◽

Solution Methodology

AbstractA new solution methodology is proposed for solving efficiently Helmholtz problems. The proposed method falls in the category of the discontinuous Galerkin methods. However, unlike the existing solution methodologies, this method requires solving (a) well-posed local problems to determine the primal variable, and (b) a global positive semi-definite Hermitian system to evaluate the Lagrange multiplier needed to restore the continuity across the element edges. Illustrative numerical results obtained for two-dimensional interior Helmholtz problems are presented to assess the accuracy and the stability of the proposed solution methodology.

L2stability analysis of the central discontinuous Galerkin method and a comparison between the central and regular discontinuous Galerkin methods

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an:2008018 ◽

2008 ◽

Vol 42 (4) ◽

pp. 593-607 ◽

Cited By ~ 49

Author(s):

Yingjie Liu ◽

Chi-Wang Shu ◽

Eitan Tadmor ◽

Mengping Zhang

Keyword(s):

Galerkin Method ◽

Discontinuous Galerkin ◽

Galerkin Methods ◽

Central Discontinuous Galerkin

A Weakly Penalized Discontinuous Galerkin Method for Radiation in Dense, Scattering Media

Computational Methods in Applied Mathematics ◽

10.1515/cmam-2016-0023 ◽

2016 ◽

Vol 16 (4) ◽

pp. 563-577

Author(s):

Guido Kanschat ◽

José Pablo Lucero Lorca

Keyword(s):

Asymptotic Analysis ◽

Galerkin Method ◽

Discontinuous Galerkin ◽

Radiation Transport ◽

Isotropic Scattering ◽

Galerkin Methods ◽

Penalty Parameter ◽

Scattering Media

AbstractWe review the derivation of weakly penalized discontinuous Galerkin methods for scattering dominated radiation transport and extend the asymptotic analysis to non-isotropic scattering. We focus on the influence of the penalty parameter on the edges and derive a new penalty for interior edges and boundary fluxes. We study how the choice of the penalty parameters influences discretization accuracy and solver speed.

Stability and Boundary Resolution Analysis of the Discontinuous Galerkin Method Applied to the Reynolds-Averaged Navier--Stokes Equations Using the Spalart--Allmaras Model

SIAM Journal on Scientific Computing ◽

10.1137/110834615 ◽

2013 ◽

Vol 35 (3) ◽

pp. B666-B700 ◽

Cited By ~ 5

Author(s):

Marcus Drosson ◽

Koen Hillewaert ◽

Jean-André Essers

Keyword(s):

Galerkin Method ◽

Discontinuous Galerkin ◽

Stokes Equations ◽

Navier Stokes ◽

Navier Stokes Equations ◽

Resolution Analysis ◽

Reynolds Averaged Navier Stokes

A p-robust polygonal discontinuous Galerkin method with minus one stabilization

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202521500597 ◽

2021 ◽

pp. 1-37

Author(s):

Silvia Bertoluzza ◽

Ilaria Perugia ◽

Daniele Prada

Keyword(s):

Error Analysis ◽

Discontinuous Galerkin ◽

Numerical Experiments ◽

Convergence Rates ◽

Galerkin Methods ◽

Poisson Problem ◽

Optimal Convergence Rates ◽

Stability And Error Analysis

In this paper, we introduce a new stabilization for discontinuous Galerkin methods for the Poisson problem on polygonal meshes, which induces optimal convergence rates in the polynomial approximation degree [Formula: see text]. The stabilization is obtained by penalizing, in each mesh element [Formula: see text], a residual in the norm of the dual of [Formula: see text]. This negative norm is algebraically realized via the introduction of new auxiliary spaces. We carry out a [Formula: see text]-explicit stability and error analysis, proving [Formula: see text]-robustness of the overall method. The theoretical findings are demonstrated in a series of numerical experiments.

Modeling Magma Dynamics with a Mixed Fourier Collocation — Discontinuous Galerkin Method

Communications in Computational Physics ◽

10.4208/cicp.030210.240910a ◽

2011 ◽

Vol 10 (2) ◽

pp. 433-452 ◽

Cited By ~ 7

Author(s):

Alan R. Schiemenz ◽

Marc A. Hesse ◽

Jan S. Hesthaven

Keyword(s):

Galerkin Method ◽

Discontinuous Galerkin ◽

Hyperbolic Equations ◽

Type System ◽

High Order ◽

Galerkin Methods ◽

Magma Dynamics ◽

Efficient Representation ◽

Fourier Collocation

AbstractA high-order discretization consisting of a tensor product of the Fourier collocation and discontinuous Galerkin methods is presented for numerical modeling of magma dynamics. The physical model is an advection-reaction type system consisting of two hyperbolic equations and one elliptic equation. The high-order solution basis allows for accurate and efficient representation of compaction-dissolution waves that are predicted from linear theory. The discontinuous Galerkin method provides a robust and efficient solution to the eigenvalue problem formed by linear stability analysis of the physical system. New insights into the processes of melt generation and segregation, such as melt channel bifurcation, are revealed from two-dimensional time-dependent simulations.