Symmetric Interior Penalty Galerkin Research Articles

A modified symplectic discontinuous Galerkin method for acoustic and elastic wave simulations

Numerically solving seismic wave equations is vital to large-scale forward modeling and full waveform inversion. In this paper, a new modified symplectic discontinuous Galerkin (MSDG) method is proposed to solve the acoustic and elastic equations. The MSDG method employs a symmetric interior penalty Galerkin formulation as the space discretization. The time discretization is based on a modified symplectic partitioned Runge–Kutta scheme with minimized phase error. Thus, the MSDG method has the advantages of high accuracy, being flexible to deal with complex geometric boundaries and internal structures, and stable for long time simulations. The numerical stability conditions, dispersion and dissipation are investigated in detail for the MSDG method. To validate the method, we carry out several numerical examples for solving the acoustic and elastic wave equations in various media. The numerical results show that the MSDG method can effectively suppress the numerical dispersion and is suitable for wavefield simulations.

A Priori Analysis of a Symmetric Interior Penalty Discontinuous Galerkin Finite Element Method for a Dynamic Linear Viscoelasticity Model

Abstract The stress-strain constitutive law for viscoelastic materials such as soft tissues, metals at high temperature, and polymers can be written as a Volterra integral equation of the second kind with a fading memory kernel. This integral relationship yields current stress for a given strain history and can be used in the momentum balance law to derive a mathematical model for the resulting deformation. We consider such a dynamic linear viscoelastic model problem resulting from using a Dirichlet–Prony series of decaying exponentials to provide the fading memory in the Volterra kernel. We introduce two types of internal variable to replace the Volterra integral with a system of auxiliary ordinary differential equations and then use a spatially discontinuous symmetric interior penalty Galerkin (SIPG) finite element method and – in time – a Crank–Nicolson method to formulate the fully discrete problems: one for each type of internal variable. We present a priori stability and error analyses without using Grönwall’s inequality and with the result that the constants in our estimates grow linearly with time rather than exponentially. In this sense, the schemes are therefore suited to simulating long time viscoelastic response, and this (to our knowledge) is the first time that such high quality estimates have been presented for SIPG finite element approximation of dynamic viscoelasticity problems. We also carry out a number of numerical experiments using the FEniCS environment (https://fenicsproject.org), describe a simulation using “real” material data, and explain how the codes can be obtained and all of the results reproduced.

Two reduced non-smooth Newton's methods for discretised state constrained optimal control problem governed by advection-diffusion equation

We are interested in solving numerically an optimal control problem governed by advection-diffusion equation with state inequality constraints. This problem can be implemented to modelise the minimisation of air pollution. First the infinite dimensional problem is discretised by application of the upwind symmetric interior penalty galerkin (SIPG) method. Then, we write the optimality conditions for the discretised problem. We reformulate these conditions into an equivalent system of nonlinear and non-smooth equations by the use of the Fischer-Burmeister function. Then, two effective algorithms that combine the non-smooth Newton's method and an active (index) set technique or pseudo inverse method are applied to solve this system. At the end, numerical examples are presented to illustrate the performance of our methods.

A combined stabilized mixed finite element and discontinuous Galerkin method for coupled Stokes and Darcy flows with transport

This paper presents a combined stabilized mixed finite element and discontinuous Galerkin method for coupled Stokes-Darcy flows model with transport, where the fluid viscosity depends on the concentration. We use nonconforming piecewise linear Crouzeix-Raviart (C-R) element to approximate velocity, piecewise constant function to approximate pressure and the symmetric interior penalty Galerkin (SIPG) method to solve concentration equation. A “cut-off” operator is introduced into SIPG scheme to avoid the assumption on the boundness of infinity norms of approximate velocity in convergence analysis. Optimal a priori error estimates for the full discrete scheme are obtained. Finally, some numerical examples are presented to verify the theoretical analysis, and a water injection oil production process is simulated to illustrate the practicability of our method.

Space-Time Petrov-Discontinuous Galerkin Finite Element Method for Solving Linear Convection-Diffusion Problems

The paper presents the theory of the space-time Petrov-discontinuous Galerkin finite element (PDGFE) method for the discretization of the nonstationary linear convection-diffusion problems. The PDGFE method is modified for the discontinuous Galerkin finite element (DGFE) method in the case of the symmetric interior penalty Galerkin (SIPG) scheme. PDGFE method is applied separately in space using different space gride on different time levels. We prove the properties of the bilinear form aPD, m (u, ν) (V − elliptic and continuity) stability and prove the approximate solution converges with the error of order o (h 2 + τ 3). A numerical experiment is carried out to confirm the theoretical conclusions.

The SIPG method of Dirichlet boundary optimal control problems with weakly imposed boundary conditions

<abstract><p>In this paper, we consider the symmetric interior penalty Galerkin (SIPG) method which is one of Discontinuous Galerkin Methods for the Dirichlet optimal control problems governed by linear advection-diffusion-reaction equation on a convex polygonal domain and the difficulties which we faced while solving this problem numerically. Since standard Galerkin methods have failed when the boundary layers have occurred and advection diffusion has dominated, these difficulties can occur in the cases of higher order elements and non smooth Dirichlet data in using standard finite elements. We find the most convenient natural setting of Dirichlet boundary control problem for the Laplacian and the advection diffusion reaction equations.After converting the continuous problem to an optimization problem, we solve it by "discretize-then-optimize" approach. In final, we estimate the optimal priori error estimates in suitable norms of the solutions and then support the result and the features of the method with numerical examples on the different kinds of domain.</p></abstract>

A strongly conservative finite element method for the coupled Stokes and dual-porosity model

In this work, we propose a numerical finite element discretization with strong mass conservation for the coupled Stokes and dual-porosity model. Based on divergence-conforming finite element spaces and piecewise discontinuous finite element spaces, this strongly conservative discretization is constructed by utilizing the symmetric interior penalty Galerkin method and mixed finite element method to discrete the governing equations of Stokes region and dual-porosity domain, respectively. In light of a discrete inf–sup condition, we present the well-posedness of discrete scheme and prove priori error estimates. After using uniformly matching meshes and the lower order finite element spaces of velocity and pressure, some numerical examples are given to validate the analysis of convergence and strong mass conservation. Further, these numerical results support our findings and illustrate the applicability of the coupled Stokes–dual-porosity model.

Numerical analysis of the neutron multigroupSPNequations

The multigroup neutron SP N equations, which are an approximation of the neutron transport equation, are used to model nuclear reactor cores. In their steady state, these equations can be written as a source problem or an eigenvalue problem. We study the resolution of those two problems with an H 1 -conforming finite element method and a Discontinuous Galerkin method, namely the Symmetric Interior Penalty Galerkin method.

An adaptively enriched coarse space for Schwarz preconditioners for P1 discontinuous Galerkin multiscale finite element problems

Abstract In this paper, we propose a two-level additive Schwarz domain decomposition preconditioner for the symmetric interior penalty Galerkin method for a second-order elliptic boundary value problem with highly heterogeneous coefficients. A specific feature of this preconditioner is that it is based on adaptively enriching its coarse space with functions created by solving generalized eigenvalue problems on thin patches covering the subdomain interfaces. It is shown that the condition number of the underlined preconditioned system is independent of the contrast if an adequate number of functions are used to enrich the coarse space. Numerical results are provided to confirm this claim.

Two-Level Schwarz Methods for a Discontinuous Galerkin Approximation of Elliptic Problems with Jump Coefficients

We present two-level nonoverlapping and overlapping Schwarz preconditioners for the linear algebraic system arising from the weighted symmetric interior penalty Galerkin approximation of elliptic problems with highly heterogeneous coefficients. The coarse space is constructed by the local Dirichlet-to-Neumann maps the theoretical results show that the condition number of the preconditioned system is independent of the discontinuous coefficient, the number of subdomains and the mesh size for the nonoverlapping case. For the overlapping case adding an extra assumption of coefficient distribution, the similar conclusion is also obtained. Numerical experiments validate the theoretical results and illustrate the performance and robustness of the proposed two-level methods.

Three-dimensional multi-patch isogeometric analysis of composite laminates with a discontinuous Galerkin approach

In this study, a three-dimensional discontinuous Galerkin isogeometric analysis framework is presented for the analysis of composite laminates. Non-uniform rational B-splines are employed as basis functions for both geometric and computational implementations. From a practical point of view, modeling with multiple non-uniform rational B-spline patches is required in many different applications due to the complexity of computational domains. Then, a special numerical technique is necessary to couple different non-uniform rational B-spline patches to carry out the isogeometric analysis. In this study, therefore, one of the discontinuous Galerkin methods, namely, symmetric interior penalty Galerkin formulation is utilized to deal with multi-patch isogeometric analysis applications. In order to show the applicability of the proposed framework, composite laminates under sinusoidally distributed load with different stacking sequences are studied in the numerical examples. The predicted results are compared with those obtained by the three-dimensional elasticity solutions and various numerical models available in the literature.

Analysis of penalty parameters for interior penalty Galerkin methods

PurposeThe purpose of this paper is to analyse the influence of penalty parameters for an interior penalty Galerkin method, namely, the symmetric interior penalty Galerkin method.Design/methodology/approachFirst of all, the solution of a simple model problem is computed and compared to the exact solution, which is a periodic function. Afterwards, a two-dimensional magnetostatic field problem described by the magnetic vector potential A is considered. In particular, penalty parameters depending on the polynomial degree, the properties of the elements and the material are considered. The analysis is performed by varying the polynomial degree and the mesh sizes on a structured and an unstructured mesh. Additionally, the penalty parameter is varied in a specific range.FindingsChoosing the penalty parameter correctly plays an important role as the stability and the convergence of the numerical scheme can be affected. For a structured mesh, a limiting value for the penalty parameter can be calculated beforehand, whereas for an unstructured mesh, the choice of the penalty parameter can be cumbersome.Originality/valueThis paper shows that there exist different penalty parameters which can be taken into account to solve the considered problems. One can choose a global penalty parameter to obtain a stable solution, which is a sharp estimation. There has always to be the consideration to guarantee the coercivity of the bilinear form while minimising the number of iterations.

Optimal control problem of the two-dimensional modified anomalous subdiffusion equation with discontinuous Galerkin approximation

In this paper, we consider an optimal control problem (OCP) governed by a modified anomalous subdiffusion equation with box constraints on the control. We apply space–time discontinuous Galerkin method to discretize the problem. Specifically, symmetric interior penalty Galerkin method and piecewise discontinuous constant temporal approximations are used. Then, we discuss the equivalence of optimize then discretize and discretize then optimize approaches in a general setting. We derive stability and convergence analysis for the fully discrete problem. At the end, we present some numerical results to measure the numerical rate of convergence.

On the composite discontinuous Galerkin method for simulations of electric properties of semiconductor devices

In this paper, a variant of discretization of the van Roosbroeck equations in the equilibrium state with the Composite Discontinuous Galerkin Method for the rectangular domain is discussed. It is based on Symmetric Interior Penalty Galerkin (SIPG) method. The proposed method accounts for lower regularity of the solution on the interfaces of devices' layers. It is shown that the discrete problem is well-defined and that discrete solution is unique. Error estimates are derived. Finally, numerical simulations are presented.

Comparison Between Discontinuous Galerkin Method and Cohesive Element Method: On the Convergence and Dynamic Wave Propagation Issue

The zero-thickness cohesive element method (CEM) is a powerful way to simulate crack propagation. However, it suffers from artificial compliance issue and alters the wave speed in dynamic analyses. In this article, we show that the CEM formulation can be correlated to the Babuška and Zlámal, discontinuous Galerkin formulation, and switching to symmetric interior penalty Galerkin (SIPG) formulation improves the accuracy of wave speed approximation. Moreover, we demonstrate the SIPG formulation can better control the nodal jumps between element interfaces with a lower cost than CEM. A convergence study is carried out to support this SIPG robustness.

Multipatch discontinuous Galerkin isogeometric analysis of composite laminates

This paper is concerned with the isogeometric analysis (IGA) of composite laminates under cylindrical bending. Non-uniform rational B-splines (NURBS) are employed as basis functions for both geometric and computational implementations. In order to account for multiple domains, each lamina is modeled as a single NURBS patch. This multipatch representation corresponds to decomposition of the computational domain (composite laminate) into non-overlapping subdomains. As NURBS patches are discontinuous across their boundaries, a standard FEA-like procedure does not work for multipatch IGA; an additional numerical technique is required for coupling NURBS patches. Therefore, in this paper, one of the discontinuous Galerkin (DG) methods, namely symmetric interior penalty Galerkin formulation, is employed to allow for discontinuities. For numerical calculations, a composite laminate with stacking sequences $$0^{\circ }{/}90^{\circ }$$ and $$0^{\circ }{/}90^{\circ }{/}0^{\circ }$$ , respectively, is adopted. The stresses are calculated along the thickness of the composite laminate, subjected to a sinusoidal load, and they are compared with the analytical solutions. It is shown that DG–IGA gives a better approximation in comparison with the standard IGA.

Energy Stable Discontinuous Galerkin Finite Element Method for the Allen–Cahn Equation

In this paper, we investigate numerical solution of Allen–Cahn equation with constant and degenerate mobility, and with polynomial and logarithmic energy functionals. We discretize the model equation by symmetric interior penalty Galerkin (SIPG) method in space, and by average vector field (AVF) method in time. We show that the energy stable AVF method as the time integrator for gradient systems like the Allen–Cahn equation satisfies the energy decreasing property for fully discrete scheme. Numerical results reveal that the discrete energy decreases monotonically, the phase separation and metastability phenomena can be observed, and the ripening time is detected correctly.

Symmetric interior penalty Galerkin method for fractional-in-space phase-field equations

Fractional differential equations are becoming increasingly popular as a modelling tool todescribe a wide range of non-classical phenomena with spatial heterogeneities throughout the appliedsciences and engineering. However, the non-local nature of the fractional operators causes essentialdifficulties and challenges for numerical approximations. We here investigate the numerical solution offractional-in-space phase-field models such as Allen-Cahn and Cahn-Hilliard equations via the contourintegral method (CIM) for computing the fractional power of a matrix times a vector. Time discretizationis performed by the first-and second-order implicit-explicit schemes with an adaptive time-stepsize approach, whereas spatial discretization is performed by a symmetric interior penalty Galerkin(SIPG) method. Several numerical examples are presented to illustrate the effect of the fractionalpower.

Adaptive discontinuous Galerkin approximation of optimal control problems governed by transient convection-diffusion equations

In this paper, we investigate a posteriori error estimates of a control-constrained optimal control problem governed by a time-dependent convection diffusion equation. The control constraints are handled by using the primal-dual active set algorithm as a semi-smooth Newton method and by adding a Moreau-Yosida-type penalty function to the cost functional. Residual-based error estimators are proposed for both approaches. The derived error estimators are used as error indicators to guide the mesh refinements. A symmetric interior penalty Galerkin method in space and a backward Euler method in time are applied in order to discretize the optimization problem. Numerical results are presented, which illustrate the performance of the proposed error estimators.

Discontinuous Galerkin gradient discretisations for the approximation of second-order differential operators in divergence form

We include in the gradient discretisation method (GDM) framework two numerical schemes based on discontinuous Galerkin approximations: the symmetric interior penalty Galerkin (SIPG) method, and the scheme obtained by averaging the jumps in the SIPG method. We prove that these schemes meet the main mathematical gradient discretisation properties on any kind of polytopal mesh, by adapting discrete functional analysis properties to our precise geometrical hypotheses. Therefore, these schemes inherit the general convergence properties of the GDM, which hold for instance in the cases of the p-Laplace problem and of the anisotropic and heterogeneous diffusion problem. This is illustrated by simple 1D and 2D numerical examples.