Interior Penalty Discontinuous Galerkin Method Research Articles

A Space-Time Interior Penalty Discontinuous Galerkin Method for the Wave Equation

A new higher-order accurate space-time discontinuous Galerkin (DG) method using the interior penalty flux and discontinuous basis functions, both in space and in time, is presented and fully analyzed for the second-order scalar wave equation. Special attention is given to the definition of the numerical fluxes since they are crucial for the stability and accuracy of the space-time DG method. The theoretical analysis shows that the DG discretization is stable and converges in a DG-norm on general unstructured and locally refined meshes, including local refinement in time. The space-time interior penalty DG discretization does not have a CFL-type restriction for stability. Optimal order of accuracy is obtained in the DG-norm if the mesh size h and the time step Delta t satisfy hcong CDelta t, with C a positive constant. The optimal order of accuracy of the space-time DG discretization in the DG-norm is confirmed by calculations on several model problems. These calculations also show that for pth-order tensor product basis functions the convergence rate in the L^infty and L^2-norms is order p+1 for polynomial orders p=1 and p=3 and order p for polynomial order p=2.

A new DG method for a pure–stress formulation of the Brinkman problem with strong symmetry

<p style='text-indent:20px;'>A strongly symmetric stress approximation is proposed for the Brinkman equations with mixed boundary conditions. The resulting formulation solves for the Cauchy stress using a symmetric interior penalty discontinuous Galerkin method. Pressure and velocity are readily post-processed from stress, and a second post-process is shown to produce exactly divergence-free discrete velocities. We demonstrate the stability of the method with respect to a DG-energy norm and obtain error estimates that are explicit with respect to the coefficients of the problem. We derive optimal rates of convergence for the stress and for the post-processed variables. Moreover, under appropriate assumptions on the mesh, we prove optimal <inline-formula><tex-math id="M1">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-error estimates for the stress. Finally, we provide numerical examples in 2D and 3D.</p>

An equilibrated a posteriori error estimator for an Interior Penalty Discontinuous Galerkin approximation of the p-Laplace problem

Abstract We are concerned with an Interior Penalty Discontinuous Galerkin (IPDG) approximation of the p-Laplace equation and an equilibrated a posteriori error estimator. The IPDG method can be derived from a discretization of the associated minimization problem involving appropriately defined reconstruction operators. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W 1,p norm and relies on the construction of an equilibrated flux in terms of a numerical flux function associated with the mixed formulation of the IPDG approximation. The relationship with a residual-type a posteriori error estimator is established as well. Numerical results illustrate the performance of both estimators.

Two-level iterative algorithms for two-dimensional H(curl)-elliptic problem with a symmetry interior penalty discontinuous Galerkin method

In this paper, we first present a symmetry interior penalty discontinuous Galerkin (SIPDG) discretization for two-dimensional H(curl)-elliptic problem, then develop the corresponding two-level iterative algorithms. The main idea of the construction of algorithms is to decompose the DG finite element spaces into the high frequency component and edge finite element spaces. We also present some numerical experiments to show the efficiency of our algorithms.

Generalized multiscale discontinuous Galerkin method for convection–diffusion equation in perforated media

In this work, we present a Generalized Multiscale Discontinuous Galerkin Method (GMsDGM) for the convection–diffusion equation in perforated media. In order to numerically solve the problem on the fine grid and use the solution as a reference solution, we construct the grid that resolves perforation on the fine grid level and performs finite element approximation using the Interior Penalty Discontinuous Galerkin method (IPDG). To reduce the size of the fine grid system, we present the construction of the coarse grid approximation using a Generalized Multiscale Discontinuous Galerkin Method (GMsDGM). In GMsDGM, we construct the multiscale basis functions in the local domains by solutions of local convection–diffusion problems with different boundary conditions and performing dimension reduction by solutions of local spectral problems. We investigate several types of multiscale space constructions based on the various choices of boundary conditions of the local domains. Numerical results are presented for several test problems with different velocities fields and perforations distributions. We also investigate the influence of various constructions of multiscale basis functions with a different number of basis functions and different model parameters.

Performances of Hybridized-, Embedded-, and Weighted-Interior Penalty Discontinuous Galerkin Methods for Heterogeneous and Anisotropic Diffusion Problems

The present paper discusses families of Interior Penalty Discontinuous Galerkin (IP) methods for solving heterogeneous and anisotropic diffusion problems. Specifically, we focus on distinctive schemes, namely the Hybridized-, Embedded-, and Weighted-IP schemes, leading to final matrixes of different sizes and sparsities. Both the Hybridized- and Embedded-IP schemes are eligible for static condensation, and their degrees of freedom are distributed on the mesh skeleton. In contrast, the unknowns are located inside the mesh elements for the Weighted-IP variant. For a given mesh, it is well-known that the number of degrees of freedom related to the standard Discontinuous Galerkin methods increases more rapidly than those of the skeletal approaches (Hybridized- and Embedded-IP). We then quantify the impact of the static condensation procedure on the computational performances of the different IP classes in terms of robustness, accuracy, and CPU time. To this aim, numerical experiments are investigated by considering strong heterogeneities and anisotropies. We analyze the fixed error tolerance versus the run time and mesh size to guide our performance criterion. We also outlined some relationships between these Interior Penalty schemes. Eventually, we confirm the superiority of the Hybridized- and Embedded-IP schemes, regardless of the mesh, the polynomial degree, and the physical properties (homogeneous, heterogeneous, and/or anisotropic).

Discretization error estimates for discontinuous Galerkin isogeometric analysis

Isogeometric analysis is a spline-based discretization method to partial differential equations which show the approximation power of a high-order method. The number of degrees of freedom, however, is as small as the number of degrees of freedom of a low-order method. This does not come for free as the original formulation of isogeometric analysis requires a global geometry function. Since this is too restrictive for many kinds of applications, the domain is usually decomposed into patches, where each patch is parameterized with its own geometry function. In simpler cases, the patches can be combined in a conforming way. However, for non-matching discretizations or for varying coefficients, a non-conforming discretization is desired. An symmetric interior penalty discontinuous Galerkin method for isogeometric analysis has been previously introduced. In the present paper, we give error estimates that are explicit in the spline degree. This opens the door towards the construction and the analysis of fast linear solvers, particularly multigrid solvers for non-conforming multipatch isogeometric analysis.

Numerical error analysis for an energy-stable HDG method for the Allen–Cahn equation

This paper presents an energy-stable hybridizable interior penalty discontinuous Galerkin method for the Allen–Cahn equation. To obtain an unconditionally energy stable scheme, the energy potential is split into a sum of a convex and concave function. Energy stability for the proposed scheme is proven to hold for arbitrary time. Existence and uniqueness for the scheme is also established. Under standard assumptions on the energy potential (Lipschitz continuity), we demonstrate rigorously that the method converges optimally for symmetric schemes, and suboptimally for nonsymmetric schemes. Several examples are provided which numerically verify and validate the method.

High-fidelity analysis of multilayered shells with cut-outs via the discontinuous Galerkin method

A novel numerical method for the analysis of multilayered shells with cut-outs is presented. In the proposed approach, the shell geometry is represented via either analytical functions or NURBS parametrizations, while generally-shaped cut-outs are defined implicitly within the shell modelling domain via a level set function. The multilayered shell problem is addressed via the Equivalent-Single-Layer approach whereby high-order polynomial functions are employed to approximate the covariant components of the displacement field throughout the shell thickness. The shell governing equations are then derived from the Principle of Virtual Displacements of three-dimensional elasticity and solved via an Interior Penalty discontinuous Galerkin method over a discretization of the shell modelling domain that is obtained by intersecting a background structured grid with the level set function defining the cut-outs. To maintain high-order accuracy even in proximity of the embedded cut-outs, high-order accurate quadrature rules for implicitly-defined regions are employed to compute the integrals of the method while a cell-merging technique avoids the presence of overly small cut cells. The combined use of these features represents the novelty of the proposed method and provides a high-fidelity approach to the analysis of multilayered shells with cut-outs. Numerical tests are performed to model the static response of a cylindrical shell and a NURBS-based shell with a cut-out. The obtained results are compared with those obtained using the Finite Element method and show the accuracy and the computational efficiency of method.

ℎ𝑝-version discontinuous Galerkin methods on essentially arbitrarily-shaped elements

We extend the applicability of the popular interior penalty discontinuous Galerkin method discretizing advection-diffusion-reaction problems to meshes comprising extremely general, essentially arbitrarily-shaped element shapes. In particular, our analysis allows for curved element shapes, without the use of non-linear elemental maps. The feasibility of the method relies on the definition of a suitable choice of the discontinuity penalization, which turns out to be explicitly dependent on the particular element shape, but essentially independent on small shape variations. This is achieved upon proving extensions of classical trace and Markov-type inverse estimates to arbitrary element shapes. A further new H 1 − L 2 H^1-L_2 -type inverse estimate on essentially arbitrary element shapes enables the proof of inf-sup stability of the method in a streamline-diffusion-like norm. These inverse estimates may be of independent interest. A priori error bounds for the resulting method are given under very mild structural assumptions restricting the magnitude of the local curvature of element boundaries. Numerical experiments are also presented, indicating the practicality of the proposed approach.

Error estimates for a class of discontinuous Galerkin methods for nonsmooth problems via convex duality relations

We devise and analyze a class of interior penalty discontinuous Galerkin methods for nonlinear and nonsmooth variational problems. Discrete duality relations are derived that lead to optimal error estimates in the case of total-variation regularized minimization or obstacle problems. The analysis provides explicit estimates that precisely determine the role of stabilization parameters. Numerical experiments suppport the optimality of the estimates.

Conservative numerical methods for the reinterpreted discrete fracture model on non-conforming meshes and their applications in contaminant transportation in fractured porous media

The discrete fracture model (DFM) has been widely used to simulate fluid flow in fractured porous media. Traditional DFM is considered to be limited on conforming meshes, hence significant difficulty may arise in generating high-quality unstructured meshes due to the complexity of the fracture networks. Recently, Xu and Yang reinterpreted DFM and demonstrated that it can actually be extended to non-conforming meshes without any essential changes. However, the continuous Galerkin (CG) method was applied and the local mass conservation was missing. This paper is a follow-up work, and we apply the interior penalty discontinuous Galerkin (IPDG) method and enriched Galerkin (EG) method for the pressure equation. With the numerical fluxes, the local mass is conservative. As an application, we combine the reinterpreted DFM (RDFM) with the incompressible miscible displacements in porous media. The bound-preserving techniques are applied to the coupled system. We can theoretically guarantee that the concentration is between 0 and 1. Finally, several numerical experiments are given to demonstrate the good performance of the RDFM based on the above two methods on non-conforming meshes and the effectiveness of the bound-preserving technique.

An Edge Multiscale Interior Penalty Discontinuous Galerkin method for heterogeneous Helmholtz problems with large varying wavenumber

We propose an Edge Multiscale Finite Element Method (EMsFEM) based on an Interior Penalty Discontinuous Galerkin (IPDG) formulation for the heterogeneous Helmholtz problems with large wavenumber. A novel local multiscale space is constructed by solving local problems with a mixed boundary condition composed of a nonhomogeneous Dirichlet boundary condition and an absorbing boundary condition, which can capture the local behavior of the wave propagation and local media information. The key ingredient of our method consists of choosing appropriate Dirichlet data inspired by recent development on edge multiscale basis functions [9,24,39,49]. An IPDG formulation is applied to facilitate generating a sparse linear system and to reduce computational complexity. The convergence rate is derived for wavelet-based and polynomial-based edge multiscale basis functions. Extensive numerical tests in two and three dimensional heterogeneous media are presented to show the supreme performance of our method.

Residual-Based A Posteriori Error Estimates for $hp$-Discontinuous Galerkin Discretizations of the Biharmonic Problem

We introduce a residual-based a posteriori error estimator for a novel hp-version interior penalty discontinuous Galerkin method for the biharmonic problem in two and three dimensions. We prove that the error estimate provides an upper bound and a local lower bound on the error and that the lower bound is robust to the local mesh size but not the local polynomial degree. The suboptimality in terms of the polynomial degree is fully explicit and grows at most algebraically. Our analysis does not require the existence of a C1-conforming piecewise polynomial space and is instead based on an elliptic reconstruction of the discrete solution to the H2 space and a generalised Helmholtz decomposition of the error. This is the first hp-version error estimator for the biharmonic problem in two and three dimensions. The practical behaviour of the estimator is investigated through numerical examples in two and three dimensions.

Layer-Wise Discontinuous Galerkin Methods for Piezoelectric Laminates

In this work, a novel high-order formulation for multilayered piezoelectric plates based on the combination of variable-order interior penalty discontinuous Galerkin methods and general layer-wise plate theories is presented, implemented and tested. The key feature of the formulation is the possibility to tune the order of the basis functions in both the in-plane approximation and the through-the-thickness expansion of the primary variables, namely displacements and electric potential. The results obtained from the application to the considered test cases show accuracy and robustness, thus confirming the developed technique as a supplementary computational tool for the analysis and design of smart laminated devices.

Quantum element method for quantum eigenvalue problems derived from projection-based model order reduction

An effective multi-element simulation methodology for quantum eigenvalue problems is investigated. The approach is derived from a reduced-order model based on a data-driven learning algorithm, together with the concept of domain decomposition. The approach partitions the simulation domain of a quantum eigenvalue problem into smaller subdomains that, referred to as elements, could be the building blocks for quantum structures of interest. In this quantum element method (QEM), each element is projected onto a functional space represented by a set of basis functions (or modes) that are generated from proper orthogonal decomposition (POD). To construct a POD model for a large domain, these projected elements can be combined together, and the interior penalty discontinuous Galerkin method is applied to achieve the interface continuity and stabilize the numerical solution. The POD is able to optimize the basis functions specifically tailored to the geometry and parametric variations of the problem and can therefore substantially reduce the degree of freedom (DoF) needed to solve the Schrödinger equation. To understand the fundamental issues of the QEM, demonstrations in this study focus on examining the accuracy and DoF of the QEM influenced by the training settings for generation of POD modes, selection of the penalty number, suppression of interface discontinuities, structure size and complexity, etc. It has been shown that the QEM is able to achieve a substantial reduction in the DoF with a high accuracy even beyond the training conditions for the POD modes if the penalty number is selected within an appropriate range.

Two-grid IPDG discretization scheme for nonlinear elliptic PDEs

Two-grid interior penalty discontinuous Galerkin (IPDG) method for the mildly nonlinear second-order elliptic partial differential equations is studied in this paper. The IPDG finite element discretizations are developed and the corresponding well-posedness is established by introducing the equivalent weak formulation of IPDG method and combining Brouwer’s fixed point theorem. Some priori error estimates for discrete solution in the broken H1-norm, L2-norm and L∞-norm are derived, respectively. Two-grid method is designed for solving IPDG discretization scheme and the corresponding error estimate is provided. Numerical experiments are also shown to confirm the efficiency of the proposed approach.

A Discontinuous Galerkin Method for the Coupled Stokes and Darcy Problem

Combining the mixed discontinuous Galerkin method for the Darcy flow and the interior penalty discontinuous Galerkin methods for the Stokes problem, a locally conservative discrete scheme is proposed for numerically solving the coupled Stokes and Darcy problem. We prove the well-posedness of the solution of the proposed numerical scheme by boundedness, K-ellipticity and a discrete inf-sup condition. A priori error estimates, in proper norms are derived, and to verify the theoretical analysis, some numerical experiments are given.

A fully interior penalty discontinuous Galerkin method for variable density groundwater flow problems

Discontinuous Galerkin (DG) methods due to their robustness properties, e.g. local conservation, low numerical dispersion, and well-capturing strong shocks and physical discontinuities, are well-suited for the simulation of Variable Density Flow (VDF) in porous media. This paper aims at introducing, in a unified format, the general class of Interior Penalty DG (IPDG) methods to solve the VDF equations. A combination of symmetric, non-symmetric and incomplete IPDG methods is used to discretize both head and concentration variables. Compatibility analysis is performed to prevent the loss of accuracy of the IPDG methods in simulations of coupled flow and transport equations. An accurate technique is used for time integration, based on a non-iterative procedure and adaptive time stepping with embedded error control. Several benchmarks are investigated to validate the proposed DG scheme and to examine its performance in simulating VDF problems. The new DG scheme reproduces better the experimental data than the conventional SEAWAT model. Its results are in excellent agreement with a recent semi-analytical solution of the Henry problem, dealing with seawater intrusion under convection-dominating conditions. The performance of the DG scheme is examined by simulating the challenging problem of natural convection in porous enclosure. The method is compared against a finite element solution obtained with COMSOL multi-physics. The numerical experiments indicate clearly that high-order DG method is much more appropriate than standard conforming Galerkin method in simulating VDF problems while at the same time, guaranteeing a better precision and high-fidelity solutions. The proposed numerical method can be extended to 3D problems.

An Adaptive Multiresolution Interior Penalty Discontinuous Galerkin Method for Wave Equations in Second Order Form

In this paper, we propose a class of adaptive multiresolution (also called adaptive sparse grid) discontinuous Galerkin (DG) methods for simulating scalar wave equations in second order form in space. The two key ingredients of the schemes include an interior penalty DG formulation in the adaptive function space and two classes of multiwavelets for achieving multiresolution. In particular, the orthonormal Alpert’s multiwavelets are used to express the DG solution in terms of a hierarchical structure, and the interpolatory multiwavelets are further introduced to enhance computational efficiency in the presence of variable wave speed or nonlinear source. Some theoretical results on stability and accuracy of the proposed method are presented. Benchmark numerical tests in 2D and 3D are provided to validate the performance of the method.