Interior Penalty Discontinuous Galerkin Research Articles

Interior Penalty DGTD Method for Solving Wave Equation in Dispersive Media Described With GDM Model

In this communication, a paralleled interior penalty discontinuous Galerkin time-domain method based on the vector wave equation (IPDG-WE) combining with a generalized dispersive material (GDM) model is introduced. In addition, the auxiliary differential equation (ADE) technique is employed to establish a universal solution for scattering analysis of dispersive structures in a wide frequency range. The proposed method presents great characteristics of energy conservation and the optimal convergence rate even in low-order bases and exhibits promising competitiveness in CPU time and memory usage compared with the traditional Maxwell's equations-based discontinuous Galerkin time-domain (DGTD-ME) method. Numerical simulations are executed to demonstrate the accuracy and efficiency of the proposed method, which realize twice the memory and triple the CPU time reductions.

Numerical analysis of a second-order IPDGFE method for the Allen–Cahn equation and the curvature-driven geometric flow

The paper focuses on proposing and analyzing a nonlinear interior penalty discontinuous Galerkin finite element (IPDGFE) method for the Allen–Cahn equation, which is a reaction–diffusion model with a nonlinear singular perturbation arising from the phase separation process. We firstly present a fully discrete IPDGFE formulation based on the modified Crank–Nicolson scheme and a mid-point approximation of the potential term f(u). We then derive the energy-stability and the second-order-in-time error estimates for the proposed IPDGFE method under some regularity assumptions on the initial function u0. There are two key works in our paper. One is to establish a second-order-in-time and energy-stable IPDGFE scheme. The other is to use a discrete spectrum estimate to handle the midpoint of the discrete solutions um and um+1 in the nonlinear term, instead of using the standard Gronwall inequality technique, so we obtain that all our error bounds depend on the reciprocal of the perturbation parameter ϵ only in some lower polynomial order, instead of exponential order. As a nontrivial byproduct of our paper, we also analyze the convergence of the zero-level sets of fully discrete IPDGFE solutions to the curvature-driven geometric flow. Finally, numerical experiments are provided to demonstrate the good performance of our presented IPDGFE method, including the time and space error estimates of the discrete solutions, discrete energy-stability, and the convergence of numerical interfaces governed by the curvature-driven geometric flow in the classical motion and generalized motion.

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.

Discontinuous Galerkin Galerkin Differences for the Wave Equation in Second-Order Form

We develop interior penalty discontinuous Galerkin difference methods for the wave equation in second-order form. The new schemes are energy conserving or energy dissipating depending on the simple...

GPU-Accelerated Discontinuous Galerkin Methods on Polytopic Meshes

Discontinuous Galerkin (dG) methods on meshes consisting of polygonal/polyhedral (henceforth, collectively termed as \emph{polytopic}) elements have received considerable attention in recent years. Due to the physical frame basis functions used typically and the quadrature challenges involved, the matrix-assembly step for these methods is often computationally cumbersome. To address this important practical issue, this work proposes two parallel assembly implementation algorithms on CUDA-enabled graphics cards for the interior penalty dG method on polytopic meshes for various classes of linear PDE problems. We are concerned with both single GPU parallelization, as well as with implementation on distributed GPU nodes. The results included showcase almost linear scalability of the quadrature step with respect to the number of GPU-cores used since no communication is needed for the assembly step. In turn, this can justify the claim that polytopic dG methods can be implemented extremely efficiently, as any assembly computing time overhead compared to finite elements on `standard' simplicial or box-type meshes can be effectively circumvented by the proposed algorithms.

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.

Hp non-conforming a priori error analysis of an Interior Penalty Discontinuous Galerkin BEM for the Helmholtz equation

This work is concerned with the construction and the hp non-conforming a priori error analysis of a Discontinuous Galerkin DG numerical scheme applied to the hypersingular integral equation related to the Helmholtz problem in 3D. The main results of this article are an error bound in a norm suited to the problem and in the L2-norm. Those bounds are quasi-optimal for the h-convergence and the p-convergence. Various formulation choices and penalty functions are theoretically discussed. In particular we show that a penalty function of the shape h2p leads to a quasi-optimal convergence of the scheme. Some numerical experiments confirm the expected rates of convergence and the effect of the penalty function.

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.

Families of interior penalty hybridizable discontinuous Galerkin methods for second order elliptic problems

AbstractThe focus of this paper is the analysis of families of hybridizable interior penalty discontinuous Galerkin methods for second order elliptic problems. We derive a priori error estimates in the energy norm that are optimal with respect to the mesh size. Suboptimal L2-norm error estimates are proven. These results are valid in two and three dimensions. Numerical results support our theoretical findings, and we illustrate the computational cost of the method.

Convergence Analysis of H(div)-Conforming Finite Element Methods for a Nonlinear Poroelasticity Problem

In this paper, we introduce and analyze H(div)-conforming finite element methods for a nonlinear model in poroelasticity. More precisely, the flow variables are discretized by H(div)-conforming mixed finite elements, while the elastic displacement is approximated by the H(div)-conforming finite element with the interior penalty discontinuous Galerkin formulation. Optimal a priori error estimates are derived for both semidiscrete and fully discrete schemes.

A new discontinuous Galerkin mixed finite element method for compressible miscible displacement problem

A new combined discontinuous Galerkin method is proposed for compressible miscible displacement problem in porous media. Here, a splitting positive definite mixed finite element method is used for the pressure and Darcy velocity, while an interior penalty discontinuous Galerkin (IPDG) method is used for the transport equation. The stability and convergence of this algorithm are considered, and the optimal a priori error estimate in l∞(L2) for velocity, pressure and concentration are given. Finally we provide some numerical results to confirm our theoretical analysis, and simulate compressible fluid flows through homogeneous and isotropic porous media.

Analysis of an interior penalty DG method for the quad-curl problem

Abstract The quad-curl term is an essential part of the resistive magnetohydrodynamic equation and the fourth-order inverse electromagnetic scattering problem, which are both of great significance in science and engineering. It is desirable to develop efficient and practical numerical methods for the quad-curl problem. In this paper we first present some new regularity results for the quad-curl problem on Lipschitz polyhedron domains, and then propose a mixed finite element method for solving the quad-curl problem. With a novel discrete Sobolev imbedding inequality for the piecewise polynomials, we obtain stability results and derive error estimates based on a relatively low-regularity assumption of the exact solution.

A strongly conservative finite element method for the coupled Stokes–Biot Model

In this paper, based on interior penalty discontinuous Galerkin method and mixed finite element method we consider a strongly conservative discretization for the rearranged Stokes–Biot model. Theoretical analysis and numerical modeling are carried out. From the view of mathematics, we firstly present the existence and uniqueness of solution of the numerical scheme. Then, the analysis of stability and priori error estimates are derived. From the point of numerical simulation, we report the numerical examples under uniform meshes, which well validate the analysis of convergence and the strong mass conservation.

Convergence in the incompressible limit of new discontinuous Galerkin methods with general quadrilateral and hexahedral elements

Standard low-order finite elements, which perform well for problems involving compressible elastic materials, are known to under-perform when nearly incompressible materials are involved, commonly exhibiting the locking phenomenon. Interior penalty (IP) discontinuous Galerkin methods have been shown to circumvent locking when simplicial elements are used. The same IP methods, however, result in locking on meshes of quadrilaterals. The authors have shown in earlier work that under-integration of specified terms in the IP formulation eliminates the locking problem for rectangular elements. Here it is demonstrated through an extensive numerical investigation that the effect of using under-integration carries over successfully to meshes of more general quadrilateral elements, as would likely be used in practical applications, and results in accurate displacement approximations. Uniform convergence with respect to the compressibility parameter is shown numerically. Additionally, a stress approximation obtained here by postprocessing shows good convergence in the incompressible limit.

Pricing European and American options under Heston model using discontinuous Galerkin finite elements

This paper deals with pricing of European and American options, when the underlying asset price follows Heston model, via the interior penalty discontinuous Galerkin finite element method (dGFEM). The advantages of dGFEM space discretization with Rannacher smoothing as time integrator with nonsmooth initial and boundary conditions are illustrated for European vanilla options, digital call and American put options. The convection dominated Heston model for vanishing volatility is efficiently solved utilizing the adaptive dGFEM. For fast solution of the linear complementary problem of the American options, a projected successive over relaxation (PSOR) method is developed with the norm preconditioned dGFEM. We show the efficiency and accuracy of dGFEM for option pricing by conducting comparison analysis with other methods and numerical experiments.

Non-Conforming Nitsche Interfaces for Edge Elements in Curl–Curl-Type Problems

In this article, a methodology to incorporate non-conforming interfaces between several conforming mesh regions is presented for Maxwell’s curl–curl problem. The derivation starts from a general interior penalty discontinuous Galerkin formulation of the curl–curl problem and eliminates all interior jumps in the conforming parts but retains them across non-conforming interfaces. Therefore, it is possible to think of this Nitsche approach for interfaces as a specialization of discontinuous Galerkin on meshes, which are conforming nearly everywhere. The applicability of this approach is demonstrated in two numerical examples, including parameter jumps at the interface. A convergence study is performed for h-refinement, including the investigation of the penalization- (Nitsche-) parameter.