High-order Discontinuous Galerkin Finite Element Research Articles

A reconstructed discontinuous approximation on unfitted meshes to [formula omitted] and [formula omitted] interface problems

We propose a high-order discontinuous Galerkin finite element method for solving the H(div)- and H(curl)-elliptic interface problems on unfitted meshes. The vector-valued approximation space is constructed by the patch reconstruction with at most d degrees of freedom per element. The C2-smooth interface is allowed to intersect elements in a quite general setup. The patch reconstruction provides the stability near the interface naturally without any additional stabilization strategy. The method is based on the symmetric interior penalty method and the optimal and suboptimal convergence rates under the energy norm and L2 norm are derived. Numerical examples in both two and three dimensions are presented to demonstrate the accuracy of the method.

A high order discontinuous Galerkin method for the symmetric form of the anisotropic viscoelastic wave equation

We introduce a new symmetric treatment of anisotropic viscous terms in the viscoelastic wave equation. An appropriate memory variable treatment of stress-strain convolution terms, result into a symmetric system of first order linear hyperbolic partial differential equations, which we discretize using a high-order discontinuous Galerkin finite element method. The accuracy of the resulting numerical scheme is verified using convergence studies against analytical plane wave solutions and analytical solutions of the viscoelastic wave equation. Computational experiments are shown for various combinations of homogeneous and heterogeneous viscoelastic media in two and three dimensions.

Low-dissipation centred schemes for hyperbolic equations in conservative and non-conservative form

We propose a one-parameter family of low-dissipation centred numerical schemes for solving hyperbolic equations in conservative or non-conservative form, using finite volume or discontinuous Galerkin finite element methods. The new schemes spring out from the multi-dimensional FORCE method and are determined by a single parameter α≥1. Given an increasing sequence of real numbers 1≤α1<α2<…<αK, there corresponds a sequence of numerical schemes with stability restriction associated to a decreasing sequence of Courant numbers 1>c1>c2>…>cK>0 and a decreasing sequence of corresponding numerical viscosity functions d1>d2>…>dK. For a given Courant number ck≤1 there is a real number αk>0 and a corresponding stable scheme with minimal numerical viscosity dk. The proposed schemes suit very well the family of high-order discontinuous Galerkin finite element methods and the recently proposed class of ADER-TR schemes, whose orders of accuracy define decreasing sequences of Courant numbers, as the order of accuracy increases.The centred methods of this paper are stable in 2D and 3D in the frame of simultaneous updating formulae, unlike other centred methods, such as 1D FORCE, which are not. Furthermore, the schemes are highly accurate for slowly-moving waves, which is precisely the kind of waves that traditional centred methods smear disastrously. Additional features of the proposed schemes include ease of implementation and applicability to any hyperbolic system either in conservative or non-conservative form.Here, the proposed schemes are analysed and computationally assessed through a suite of test problems for a linear model system, for the Euler equations in one and two space dimensions, and for the Baer-Nunziato equations for compressible two-phase flow.

High-order discontinuous Galerkin methods for coastal hydrodynamics applications

This paper applies high-order discontinuous Galerkin finite element solutions of the shallow water equations to realistic coastal hydrodynamics problems. The goal is to obtain results on par with the accuracy of current low-order models, while significantly surpassing their performance. Accomplishing this requires developing mesh discretizations with boundary and parameter field representations that are consistent with the high-order accuracy of the numerical methods. These developments allow coarse-mesh, high-order, solutions to provide comparable accuracy to today’s state-of-the-art high-resolution, low-order coastal models at reduced computational expense.

High-order Discontinuous Galerkin methods for the elastodynamics equation on polygonal and polyhedral meshes

We propose and analyze a high-order Discontinuous Galerkin Finite Element Method for the approximate solution of wave propagation problems modeled by the elastodynamics equations on computational meshes made by polygonal and polyhedral elements. We analyze the well posedness of the resulting formulation, prove hp-version error a-priori estimates, and present a dispersion analysis, showing that polygonal meshes behave as classical simplicial/quadrilateral grids in terms of dispersion properties. The theoretical estimates are confirmed through various two-dimensional numerical verifications.

High-order approximation of chromatographic models using a nodal discontinuous Galerkin approach

A nodal high-order discontinuous Galerkin finite element (DG-FE) method is presented to solve the equilibrium-dispersive model of chromatography with arbitrary high-order accuracy in space. The method can be considered a high-order extension to the total variation diminishing (TVD) framework used by Javeed et al. (2011a,b, 2013) with an efficient quadrature-free implementation. The framework is used to simulate linear and non-linear multicomponent chromatographic systems. The results confirm arbitrary high-order accuracy and demonstrate the potential for accuracy and speed-up gains obtainable by switching from low-order methods to high-order methods. The results reproduce an analytical solution and are in excellent agreement with numerical reference solutions already published in the literature.

High-Order Output-Based Adaptive Simulations of Turbulent Flow in Two Dimensions

Output-based high-order adaptive results are presented for several benchmark two-dimensional turbulent-flow simulations. The discretization is a high-order discontinuous Galerkin finite element method, and the equations solved are compressible Navier–Stokes, Reynolds-averaged with a modified version of the Spalart–Allmaras one-equation model. Mesh refinement requirements are studied through automated output-based adaptation in which a discrete adjoint solution associated with an output (e.g., the drag coefficient) weights a fine-space residual and automatically selects the elements that need more resolution. The roles of high-order and mesh anisotropy are also investigated. Finally, differences are investigated between two mesh refinement strategies: hanging-node refinement of structured meshes versus metric-based remeshing of unstructured triangles.

Numerical modeling of double-layered piezoelectric transducer systems using a high-order discontinuous Galerkin method

A high-order discontinuous Galerkin finite element method, on triangular elements, is used to model piezoelectric systems. Non-destructive testing problems are studied. In particular, flex circuit transducer designs which can be used to obtain information about the surroundings. The focus is to look at simple transducer designs, where the system reduces to a two-dimensional plane strain problem or a one-dimensional longitudinal strain problem. Numerical modulations are compared to analytical/exact solutions obtained from a longitudinal strain model, in both time and frequency domains.

P-Multigrid solution of high-order discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

We present a p-multigrid solution algorithm for a high-order discontinuous Galerkin finite element discretization of the compressible Navier–Stokes equations. The algorithm employs an element line Jacobi smoother in which lines of elements are formed using coupling based on a p = 0 discretization of the scalar convection–diffusion equation. Fourier analysis of the two-level p-multigrid algorithm for convection–diffusion shows that element line Jacobi presents a significant improvement over element Jacobi especially for high Reynolds number flows and stretched grids. Results from inviscid and viscous test cases demonstrate optimal h p + 1 order of accuracy as well as p-independent multigrid convergence rates, at least up to p = 3. In addition, for the smooth problems considered, p-refinement outperforms h-refinement in terms of the time required to reach a desired high accuracy level.

On a cell entropy inequality for discontinuous Galerkin methods

We prove a cell entropy inequality for a class of high-order discontinuous Galerkin finite element methods approximating conservation laws, which implies convergence for the one-dimensional scalar convex case.

Propagation of error into regions of smoothness for non-linear approximations to hyperbolic equations

This paper applies high-order discontinuous Galerkin finite element solutions of the shallow water equations to realistic coastal hydrodynamics problems. The goal is to obtain results on par with the accuracy of current low-order models, while significantly surpassing their performance. Accomplishing this requires developing mesh discretizations with boundary and parameter field representations that are consistent with the high-order accuracy of the numerical methods. These developments allow coarse-mesh, high-order, solutions to provide comparable accuracy to today’s state-of-the-art high-resolution, low-order coastal models at reduced computational expense.