A discontinuous Galerkin multiscale method for convection-diffusion problems (original) (raw)
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Comptes Rendus Mathematique, 2011
Inspired by the reduced basis approach and modern numerical multiscale methods, we present a new framework for an efficient treatment of heterogeneous multiscale problems. The new approach is based on the idea of considering heterogeneous multiscale problems as parametrized partial differential equations where the parameters are smooth functions. We then construct, in an offline phase, a suitable localized reduced basis that is used in an online phase to efficiently compute approximations of the multiscale problem by means of a Discontinuous Galerkin method on a coarse grid. We present our approach for elliptic multiscale problems and discuss an a posteriori error estimate that can be used in the construction process of the localized reduced basis. Numerical experiments are given to demonstrate the efficiency of the new approach.
Computer Methods in Applied Mechanics and Engineering, 2005
We consider the Discontinuous Petrov-Galerkin method for the advection-diffusion model problem, and we investigate the application of the variational multiscale method to this formulation. We show the exact modeling of the fine scale modes at the element level for the linear case, and we discuss the approximate modeling both in the linear and in the non-linear cases. Furthermore, we highlight the existing link between this multiscale formulation and the p version of the finite element method. Numerical examples illustrate the behavior of the proposed scheme.
Multiscale Numerical Methods for Singularly Perturbed Convection-Diffusion Equations
International Journal of Computational Methods, 2004
We present an efficient and robust approach in the finite element framework for numerical solutions that exhibit multiscale behavior, with applications to singularly perturbed convection-diffusion problems. The first type of equation we study is the convection-dominated convection-diffusion equation, with periodic or random coefficients; the second type of equation is an elliptic equation with singularities due to discontinuous coefficients and non-smooth boundaries. In both cases, standard methods for purely hyperbolic or elliptic problems perform poorly due to sharp boundary and internal layers in the solution. We propose a framework in which the finite element basis functions are designed to capture the local small-scale behavior correctly. When the structure of the layers can be determined locally, we apply the multiscale finite element method, in which we solve the corresponding homogeneous equation on each element to capture the small scale features of the differential operato...
Mathematics of Computation, 2012
Discontinuous Galerkin (DG) methods exhibit "hidden accuracy" that makes superconvergence of this method an increasing popular topic to address. Previous investigations have focused on the superconvergent properties of ordinary differential equations and linear hyperbolic equations. Additionally, superconvergence of order k + 3 2 for the convection-diffusion equation that focuses on a special projection using the upwind flux was presented by Cheng and Shu. In this paper we demonstrate that it is possible to extend the smoothness-increasing accuracy-conserving (SIAC) filter for use on the multidimensional linear convection-diffusion equation in order to obtain 2k+m order of accuracy, where m depends upon the flux and takes on the values 0, 1 2 , or 1. The technique that we use to extract this hidden accuracy was initially introduced by Cockburn, Luskin, Shu, and Süli for linear hyperbolic equations and extended by Ryan et al. as a smoothness-increasing accuracy-conserving filter. We solve this convection-diffusion equation using the local discontinuous Galerkin (LDG) method and show theoretically that it is possible to obtain O(h 2k+m ) in the negative-order norm. By post-processing the LDG solution to a linear convection equation using a specially designed kernel such as the one by Cockburn et al., we can compute this same order accuracy in the L 2norm. Additionally, we present numerical studies that confirm that we can improve the LDG solution from O(h k+1 )t oO(h 2k+1 ) using alternating fluxes and that we actually obtain O(h 2k+2 ) for diffusion-dominated problems.
2018
In this paper, we propose a two-level overlapping additive Schwarz domain decomposition preconditioner for the symmetric interior penalty discontinuous Galerkin method for the second order elliptic boundary value problem with highly heterogeneous coefficients. A specific feature of this preconditioner is that it is based on adaptively enriching its coarse space with functions created through solving generalized eigenvalue problems on thin patches covering the subdomain interfaces. It is shown that the condition number of the underlined preconditioned system is independent of the contrast if an adequate number of functions are used to enrich the coarse space. Numerical results are provided to confirm this claim.
Generalized multiscale discontinuous Galerkin method for solving the heat problem with phase change
Journal of Computational and Applied Mathematics, 2018
In this work, we consider a numerical solution of a heat transfer problem with phase change in heterogeneous domains. For simulation of heat transfer processes with phase transitions, we use a classic Stefan model. Computational implementation is based on generalized multiscale discontinuous Galerkin method (GMsDGM). In this method the interior penalty discontinuous Galerkin method is used for the global coupling on a coarse grid. The main idea of these methods is to construct a small dimensional local solution space that can provide an efficient calculation on coarse grid level. We present numerical results for different geometries to demonstrate an accuracy of the method.
The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
SIAM Journal on Numerical Analysis, 1998
In this paper, we study the local discontinuous Galerkin (LDG) methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge-Kutta discontinuous Galerkin (RKDG) methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, high-order formal accuracy, and easy handling of complicated geometries for convection-dominated problems. It is proven that for scalar equations, the LDG methods are L 2 -stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods.
Efficient Operator-Coarsening Multigrid Schemes for Local Discontinuous Galerkin Methods
SIAM Journal on Scientific Computing, 2019
An efficient hp-multigrid scheme is presented for local discontinuous Galerkin (LDG) discretizations of elliptic problems, formulated around the idea of separately coarsening the underlying discrete gradient and divergence operators. We show that traditional multigrid coarsening of the primal formulation leads to poor and suboptimal multigrid performance, whereas coarsening of the flux formulation leads to optimal convergence and is equivalent to a purely geometric multigrid method. The resulting operator-coarsening schemes do not require the entire mesh hierarchy to be explicitly built, thereby obviating the need to compute quadrature rules, lifting operators, and other mesh-related quantities on coarse meshes. We show that good multigrid convergence rates are achieved in a variety of numerical tests on 2D and 3D uniform and adaptive Cartesian grids, as well as for curved domains using implicitly defined meshes and for multi-phase elliptic interface problems with complex geometry. Extension to non-LDG discretizations is briefly discussed.
Mathematics of Computation, 2007
In this paper, we uncover and study a new superconvergence property of a large class of finite element methods for one-dimensional convectiondiffusion problems. This class includes discontinuous Galerkin methods defined in terms of numerical traces, discontinuous Petrov-Galerkin methods and hybridized mixed methods. We prove that the so-called numerical traces of both variables superconverge at all the nodes of the mesh, provided that the traces are conservative, that is, provided they are single-valued. In particular, for a local discontinuous Galerkin method, we show that the superconvergence is order 2 p + 1 when polynomials of degree at most p are used. Extensive numerical results verifying our theoretical results are displayed.