Multisymplecticity of Hybridizable Discontinuous Galerkin Methods [PDF]
Foundations of Computational Mathematics, 2019 In this paper, we prove necessary and sufficient conditions for a hybridizable discontinuous Galerkin (HDG) method to satisfy a multisymplectic conservation law, when applied to a canonical Hamiltonian system of partial differential equations. We show that these conditions are satisfied by the "hybridized" versions of several of the most commonly-used ...
Robert I. McLachlan, Ari Stern
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Arbitrary Lagrangian-Eulerian hybridizable discontinuous Galerkin methods for incompressible flow with moving boundaries and interfaces [PDF]
Computer Methods in Applied Mechanics and Engineering, 2019 We present a class of Arbitrary Lagrangian–Eulerian hybridizable discontinuous Galerkin methods for the incompressible flow with moving boundaries and interfaces including two-phase flow with surface tension.
Guosheng Fu
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Convergence of the Discontinuous Galerkin Method for Discontinuous Solutions [PDF]
SIAM Journal on Numerical Analysis, 2005 The author proves convergence of a discontinuous Galerkin method for a convection-reaction equation under very weak assumptions on the coefficients. The proof is based on work of \textit{R. J. DiPerna} and \textit{P. L. Lions} [Invent. Math. 98, 511--547 (1989; Zbl 0696.34049)]. Applications include modeling the flow of incompressible immiscible fluids.
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Efficient low-order refined preconditioners for high-order matrix-free continuous and discontinuous Galerkin methods [PDF]
SIAM Journal on Scientific Computing, 2019 In this paper, we design preconditioners for the matrix-free solution of high-order continuous and discontinuous Galerkin discretizations of elliptic problems based on FEM-SEM equivalence and additive Schwarz methods. The high-order operators are applied
Will Pazner
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The Discontinuous Galerkin Method with Diffusion [PDF]
Mathematics of Computation, 1992 Let \(\Omega\subset \mathbb{R}^ 2\) be a bounded polygon and \(\alpha=(\alpha_ 1,\alpha_ 2)\) a unit vector. The author considers the following class of constant-coefficient convection-diffusion equations: (1) \(u_ \alpha-\sigma_ 1u_{xx}-\sigma_ 2u_{yy}=f\), where \((x,y)\in \Omega\), \(u_ \alpha=\alpha\cdot\bigtriangledown u\) and \(\sigma_ 1\) and \(\
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Discontinuous Galerkin methods with Trefftz approximations
Journal of Computational and Applied Mathematics, 2014 14 pages, 12 figures, preprint submitted at J Comput ...
Fritz Kretzschmar +3 more
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Approximate tensor-product preconditioners for very high order discontinuous Galerkin methods [PDF]
Journal of Computational Physics, 2017 In this paper, we develop a new tensor-product based preconditioner for discontinuous Galerkin methods with polynomial degrees higher than those typically employed.
Will Pazner, P. Persson
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Hybrid Discontinuous Galerkin Methods with Relaxed H(div)-Conformity for Incompressible Flows. Part I [PDF]
SIAM Journal on Numerical Analysis, 2017 The present work is the second part of a pair of papers, considering Hybrid Discontinuous Galerkin methods with relaxed H(div)-conformity. The first part mainly dealt with presenting a robust analysis with respect to the mesh size h and the introduction ...
P. Lederer, C. Lehrenfeld, J. Schöberl
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Convergence of adaptive discontinuous Galerkin methods
Mathematics of Computation, 2018 We develop a general convergence theory for adaptive discontinuous Galerkin methods for elliptic PDEs covering the popular SIPG, NIPG and LDG schemes as well as all practically relevant marking strategies. Another key feature of the presented result is, that it holds for penalty parameters only necessary for the standard analysis ...
Christian Kreuzer +1 more
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A Performance Comparison of Continuous and Discontinuous Galerkin Methods with Fast Multigrid Solvers [PDF]
SIAM Journal on Scientific Computing, 2016 This study presents a fair performance comparison of the continuous finite element method, the symmetric interior penalty discontinuous Galerkin method, and the hybridized discontinuous Galerkin (H...
M. Kronbichler, W. Wall
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