Spacetime discontinuous Galerkin methods for convection-diffusion equations

We have developed two new methods for solving convection-diffusion systems, with particular focus on the compressible Navier-Stokes equations. Our methods are extensions of a spacetime discontinuous Galerkin method for solving systems of hyperbolic conservation laws [3]. Following the original scheme, we use entropy variables as degrees of freedom and entropy stable numerical fluxes for the nonlinear convection term. We examine two different approaches for incorporating the diffusion term: the interior penalty method and the local discontinuous Galerkin approach. For both extensions, we can show an entropy stability result for convection-diffusion systems. Although our schemes are designed for systems, we focus on scalar convectiondiffusion equations in this contribution. This allows us to highlight our main ideas behind the stability proofs, which are the same for scalar equations and systems, in a simplified setting.     

Analysis of a semidiscrete discontinuous Galerkin scheme for compressible miscible displacement problem

Numerical approximation of the coupled system of compressible miscible displacement problem in porous media is considered in this paper. A continuous in time discontinuous Galerkin scheme is developed. The symmetric interior penalty discontinuous Galerkin method is used to solve both the flow and transport equations. Upwind technique is used to treat the convection term in the transport equation. The hp-a priori error bounds are derived.     

Analysis of the parareal approach based on discontinuous Galerkin method for time-dependent Stokes equations

This paper analyzes a parareal approach based on discontinuous Galerkin (DG) method for the time-dependent Stokes equations. A class of primal discontinuous Galerkin methods, namely variations of interior penalty methods, are adopted for the spatial discretization in the parareal algorithm (we call it parareal DG algorithm). We study three discontinuous Galerkin methods for the time-dependent Stokes equations, and the optimal continuous in time error estimates for the velocities and pressure are derived. Based on these error estimates, the proposed parareal DG algorithm is proved to be unconditionally stable and bounded by the error of discontinuous Galerkin discretization after a finite number of iterations. Finally, some numerical experiments are conducted which confirm our theoretical results, meanwhile, the efficiency of the parareal DG algorithm can be seen through a parallel experiment.     

Implementing a discontinuous Galerkin method for the compressible,inviscid Euler equations in the DUNE framework

A discontinuous Galerkin scheme was implemented in the DUNE framework to solve the compressible, inviscid Euler equations in a multi-dimensional Cartesian grid. It uses a HLLC Riemann solver for the numerical fluxes in the interfaces, a total variation bounded limiter to handle discontinuities, a positivity preserving limiter for near vacuum conditions, and adaptive mesh refinement (AMR). (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

DISCONTINUOUS GALERKIN METHODS AND THEIR ADAPTIVITY FOR THE TEMPERED FRACTIONAL (CONVECTION) DIFFUSION EQUATIONS

This paper focuses on the adaptive discontinuous Galerkin (DG) methods for the tempered fractional (convection) diffusion equations. The DG schemes with interior penalty for the diffusion term and numerical flux for the convection term are used to solve the equations, and the detailed stability and convergence analyses are provided. Based on the derived posteriori error estimates, the local error indicator is designed. The theoretical results and the effectiveness of the adaptive DG methods are, respectively, verified and displayed by the extensive numerical experiments. The strategy of designing adaptive schemes presented in this paper works for the general PDEs with fractional operators.     

Exponentially fitted discontinuous Galerkin schemes for singularly perturbed problems

New discontinuous Galerkin schemes in mixed form are introduced for symmetric elliptic problems of second order. They exhibit reduced connectivity with respect to the standard ones. The modifications in the choice of the approximation spaces and in the stabilization term do not spoil the error estimates. These methods are then used for designing new exponentially fitted schemes for advection dominated equations. The presented numerical tests show the good performances of the proposed schemes. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

Domain Decomposition Preconditioners for Discontinuous Galerkin Discretizations of Compressible Fluid Flows

In this article we consider the application of Schwarz-type domain decomposition preconditioners to the discontinuous Galerkin finite element approximation of the compressible Navier-Stokes equations. To discretize this system of conservation laws, we exploit the (adjoint consistent) symmetric version of the interior penalty discontinuous Galerkin finite element method. To define the necessary coarse-level solver required for the definition of the proposed preconditioner, we exploit ideas from composite finite element methods, which allow for the definition of finite element schemes on general meshes consisting of polygonal (agglomerated) elements. The practical performance of the proposed preconditioner is demonstrated for a series of viscous test cases in both two- and three-dimensions.     

On discontinuous Galerkin method and semiregular family of triangulations

Discretization of second order elliptic partial differential equations by discontinuous Galerkin method often results in numerical schemes with penalties. In this paper we analyze these penalized schemes in the context of quite general triangular meshes satisfying only a semiregularity assumption. A new (modified) penalty term is presented and theoretical properties are proven together with illustrative numerical results. This work is a part of the research project MSM 0021620839 financed by MSMT and was partly supported by the project No. 201/04/1503 of the Grant Agency of the Czech Republic.     

Discontinuous Galerkin derivative operators with applications to second‐order elliptic problems and stability

A discontinuous Galerkin (DG) finite‐element interior calculus is used as a common framework to describe various DG approximation methods for second‐order elliptic problems. Using the framework, symmetric interior‐penalty methods, local discontinuous Galerkin methods, and dual‐wind discontinuous Galerkin methods will be compared by expressing all of the methods in primal form. The penalty‐free nature of the dual‐wind discontinuous Galerkin method will be both motivated and used to better understand the analytic properties of the various DG methods. Consideration will be given to Neumann boundary conditions with numerical experiments that support the theoretical results. Many norm equivalencies will be derived laying the foundation for applying dual‐winding techniques to other problems. Copyright © 2015 John Wiley & Sons, Ltd.     

A CFL‐free explicit characteristic interior penalty scheme for linear advection‐reaction equations

We develop a CFL‐free, explicit characteristic interior penalty scheme (CHIPS) for one‐dimensional first‐order advection‐reaction equations by combining a Eulerian‐Lagrangian approach with a discontinuous Galerkin framework. The CHIPS method retains the numerical advantages of the discontinuous Galerkin methods as well as characteristic methods. An optimal‐order error estimate in the L² norm for the CHIPS method is derived and numerical experiments are presented to confirm the theoretical estimates. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

An adaptive Lagrange‐Galerkin method for the shallow‐water equations on the sphere

The shallow‐water equations on the sphere are compressible Navier‐Stokes equations that describe hydrostatic atmospheric dynamics. An adaptive Lagrange‐Galerkin method is applied on the spherical domain to develop a simplified global atmospheric model. Computational results show the experimental convergence of this numerical method.     

Space-time discontinuous Galerkin method for the solution of fluid-structure interaction

The paper is concerned with the application of the space-time discontinuous Galerkin method (STDGM) to the numerical solution of the interaction of a compressible flow and an elastic structure. The flow is described by the system of compressible Navier-Stokes equations written in the conservative form. They are coupled with the dynamic elasticity system of equations describing the deformation of the elastic body, induced by the aerodynamical force on the interface between the gas and the elastic structure. The domain occupied by the fluid depends on time. It is taken into account in the Navier-Stokes equations rewritten with the aid of the arbitrary Lagrangian-Eulerian (ALE) method. The resulting coupled system is discretized by the STDGM using piecewise polynomial approximations of the sought solution both in space and time. The developed method can be applied to the solution of the compressible flow for a wide range of Mach numbers and Reynolds numbers. For the simulation of elastic deformations two models are used: the linear elasticity model and the nonlinear neo-Hookean model. The main goal is to show the robustness and applicability of the method to the simulation of the air flow in a simplified model of human vocal tract and the flow induced vocal folds vibrations. It will also be shown that in this case the linear elasticity model is not adequate and it is necessary to apply the nonlinear model.     

Interior penalty discontinuous Galerkin methods with implicit time‐integration techniques for nonlinear parabolic equations

We prove existence and numerical stability of numerical solutions of three fully discrete interior penalty discontinuous Galerkin methods for solving nonlinear parabolic equations. Under some appropriate regularity conditions, we give the l²(H¹) and l^∞(L²) error estimates of the fully discrete symmetric interior penalty discontinuous Galerkin–scheme with the implicit θ ‐schemes in time, which include backward Euler and Crank–Nicolson finite difference approximations. Our estimates are optimal with respect to the mesh size h. The theoretical results are confirmed by some numerical experiments. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Uniqueness of numerical solutions to nonlinear parabolic equations by a fully implicit discontinuous Galerkin method

We prove uniqueness of numerical solutions to nonlinear parabolic equations approximated by a fully implicit interior penalty discontinuous Galerkin (IPDG) method, with a mesh-independent constraint on time step.     

A discontinuous Galerkin method for Stokes equation by divergence-free patch reconstruction

A discontinuous Galerkin method by patch reconstruction is proposed for Stokes flows. A locally divergence-free reconstruction space is employed as the approximation space, and the interior penalty method is adopted which imposes the normal component penalty terms to cancel out the pressure term. Consequently, the Stokes equation can be solved as an elliptic system instead of a saddle-point problem due to such weak form. The number of degree of freedoms of our method is the same as the number of elements in the mesh for different order of accuracy. The error estimations of the proposed method are given in a classical style, which are then verified by some numerical examples.     

Space–time discontinuous Galerkin finite element method for shallow water flows

A space–time discontinuous Galerkin (DG) finite element method is presented for the shallow water equations over varying bottom topography. The method results in nonlinear equations per element, which are solved locally by establishing the element communication with a numerical HLLC flux. To deal with spurious oscillations around discontinuities, we employ a dissipation operator only around discontinuities using Krivodonova's discontinuity detector. The numerical scheme is verified by comparing numerical and exact solutions, and validated against a laboratory experiment involving flow through a contraction. We conclude that the method is second order accurate in both space and time for linear polynomials.     

High-Accuracy Numerical Approximations to Several Singularly Perturbed Problems and Singular Integral Equations by Enriched Spectral Galerkin Methods

Usual spectral methods are not effective for singularly perturbed problems and singular integral equations due to the boundary layer functions or weakly singular solutions. To overcome this difficulty, the enriched spectral-Galerkin methods (ESG) are applied to deal with a class of singularly perturbed problems and singular integral equations for which the form of leading singular solutions can be determined. In particular, for easily understanding the technique of ESG, the detail of the process are provided in solving singularly perturbed problems. Ample numerical examples verify the efficiency and accuracy of the enriched spectral Galerkin methods.     

Multiscale discontinuous Petrov–Galerkin method for the multiscale elliptic problems

In this article, we present a new multiscale discontinuous Petrov–Galerkin method (MsDPGM) for multiscale elliptic problems. This method utilizes the classical oversampling multiscale basis in the framework of a Petrov–Galerkin version of the discontinuous Galerkin method, allowing us to better cope with multiscale features in the solution. MsDPGM takes advantage of the multiscale Petrov–Galerkin method (MsPGM) and the discontinuous Galerkin method (DGM). It can eliminate the resonance error completely and decrease the computational costs of assembling the stiffness matrix, thus, allowing for more efficient solution algorithms. On the basis of a new H² norm error estimate between the multiscale solution and the homogenized solution with the first‐order corrector, we give a detailed convergence analysis of the MsDPGM under the assumption of periodic oscillating coefficients. We also investigate a multiscale discontinuous Galerkin method (MsDGM) whose bilinear form is the same as that of the DGM but the approximation space is constructed from the classical oversampling multiscale basis functions. This method has not been analyzed theoretically or numerically in the literature yet. Numerical experiments are carried out on the multiscale elliptic problems with periodic and randomly generated log‐normal coefficients. Their results demonstrate the efficiency of the proposed method.     

Interior penalty discontinuous Galerkin method for Maxwell's equations: Energy norm error estimates

We develop the symmetric interior penalty discontinuous Galerkin (DG) method for the time-dependent Maxwell equations in second-order form. We derive optimal a priori error estimates in the energy norm for smooth solutions. We also consider the case of low-regularity solutions that have singularities in space.     

Error estimates for discontinuous Galerkin method for nonlinear parabolic equations

We consider the nonlinear parabolic partial differential equations. We construct a discontinuous Galerkin approximation using a penalty term and obtain an optimal L^∞(L²) error estimate.