Interior Penalty Galerkin Research Articles

An adaptively enriched coarse space for Schwarz preconditioners forP1 discontinuous Galerkin multiscale finite element problems

AbstractIn 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.

Study of the NIPG method for two–parameter singular perturbation problems on several layer adapted grids

In this paper, we apply the non-symmetric interior penalty Galerkin (NIPG) method to obtain the numerical solution of two-parameter singularly perturbed convection-diffusion-reaction boundary-value problems. In order to discretize the domain, here, we use the layer-adapted piecewise-uniform Shishkin mesh, the Bakhvalov mesh and the exponentially-graded mesh. We establish a superconvergence result of the NIPG method, that is, the proposed method is parameter-uniformly convergent with the order almost $$(k+1)$$ on the Shishkin mesh and $$(k+1)$$ on the Bakhvalov mesh and on the exponentially graded mesh in the energy norm, where k is the order of the polynomials. Numerical results comparing the three different types of meshes are presented at the end of the article supporting the theoretical error estimates.

Locking‐free interface failure modeling by a cohesive discontinuous Galerkin method for matching and nonmatching meshes

SummaryIn this work, modeling of brittle failure of the interface for a linear elastic material is presented. The idea is to integrate a novel extrinsic cohesive zone model into the incomplete interior penalty Galerkin variant of the discontinuous Galerkin (DG) method. As a result, the initial stiffness in the prefailure regime is omitted without having to remesh the crack path during the crack propagation. The interface model is used in combination with different discretization techniques, including matching and nonmatching meshes. This is possible due to the DG method's weak continuity constraint. Moreover, the locking problem in the bulk is cured by the application of a reduced Gaussian integration scheme on the boundary terms. The performance of the new cohesive discontinuous Galerkin elements with different integration schemes is compared with one of the standard intrinsic cohesive models. Due to the elimination of locking, crack initiation at the interface can be realistically displayed.

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.

On the use of reduced integration in combination with discontinuous Galerkin discretization: application to volumetric and shear locking problems

In the present work, the discontinuous Galerkin (DG) method is applied to linear elasticity for two-dimensional and three-dimensional settings. A locking-free element formulation based on reduced integration and physically-based hourglass stabilization (Q1SP) is coupled for the first time with the DG framework. The incomplete interior penalty Galerkin method is chosen, being one example of different variations of DG methods. Several 2D and 3D typical benchmark problems of linear elasticity are investigated. A selection of numerical integration schemes for the boundary terms is presented, namely reduced and mixed integration schemes. The treatment of the surface terms by means of different rules of integration shows a significant influence on the performance of the resulting DG method in combination with the standard Q1 element. This intelligent treatment of the surface part leads to a DG variant with very good convergence properties.

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.

Discontinuous Galerkin (DG) method in 3D linear elasticity with application in problems with locking

AbstractIn the present work, a three‐dimensional discontinuous Galerkin (DG) formulation is presented and investigated for linear elastic problems, with special emphasis on locking phenomena. Among different variations of DG methods, the incomplete interior penalty Galerkin (IIPG) method is applied here, where the symmetry term is absent and the approach is stable through the application of a penalty term. Different examples are calculated to investigate the locking behavior of the DG method compared to continuous Galerkin (CG) method. Additionally, A locking‐free 3D element formulation based on reduced integration and hourglass stabilization (Q1SP) is applied to compare and evaluate the results of the DG formulation in both shear and volumetric locking‐dominated problems. Due to the discontinuity of the DG elements, a new meshing approach is applied in the finite element analysis program FEAP. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Interior Penalties for Summation-by-Parts Discretizations of Linear Second-Order Differential Equations

This work focuses on multidimensional summation-by-parts (SBP) discretizations of linear elliptic operators with variable coefficients. We consider a general SBP discretization with dense simultaneous approximation terms (SATs), which serve as interior penalties to enforce boundary conditions and inter-element coupling in a weak sense. Through the analysis of adjoint consistency and stability, we present several conditions on the SAT penalties. Based on these conditions, we generalize the modified scheme of Bassi and Rebay (BR2) and the symmetric interior penalty Galerkin (SIPG) method to SBP-SAT discretizations. Numerical experiments are carried out on unstructured grids with triangular elements to verify the theoretical results.

Symmetric interior penalty Galerkin approaches for two-dimensional parabolic interface problems with low regularity solutions

This work presents novel finite element approaches for solving a parabolic partial differential equation with discontinuous coefficients and low regularity solutions in a bounded convex polyhedral domain. A spatial semi-discretization based on symmetric interior penalty Galerkin (SIPG) approximations is constructed and analyzed by using discontinuous piecewise linear functions. For smooth initial data, spatial errors in the broken L2, H1 and L2(H1) norms are proven to be optimal with respect to low regularity solutions, which are only piecewise H1+s smooth with 0<s≤1. Furthermore we present the full SIPG discretizations based on an Euler backward finite difference time discretization. The proposed approximations are shown to be unconditionally stable, and have nearly the optimal L2(L2) and L2(H1) error estimates, even when the regularities of the solutions are low on the whole domain. The error estimates are optimal with respect to time, sharply depending on the indexes in the global and local regularity. Numerical experiments for two-dimensional parabolic interface problems verify the theoretical convergence rates.

Adaptive Symmetric Interior Penalty Galerkin Method for Boundary Control Problems

We investigate an a posteriori error analysis of adaptive finite element approximations of linear-quadratic boundary optimal control problems under bilateral box constraints, which act on a Neumann boundary control. We use a symmetric interior Galerkin method as discretization technique. An efficient and reliable residual-type error estimator is introduced by invoking data oscillations. We then derive local upper and lower a posteriori error estimates for the boundary control problem. Adaptive mesh refinement indicated by a posteriori error estimates is applied. Numerical results are presented to illustrate the performance of the adaptive finite element approximation.

Discontinuous Deformation Analysis Coupling with Discontinuous Galerkin Finite Element Methods for Contact Simulations

A novel coupling scheme is presented to combine the discontinuous deformation analysis (DDA) and the interior penalty Galerkin (IPG) method for the modeling of contacts. The simultaneous equilibrium equations are assembled in a mixed strategy, where the entries are derived from both discontinuous Galerkin variational formulations and the strain energies of DDA contact springs. The contact algorithms of the DDA are generalized for element contacts, including contact detection criteria, open-close iteration, and contact submatrices. Three representative numerical examples on contact problems are conducted. Comparative investigations on the results obtained by our coupling scheme, ANSYS, and analytical theories demonstrate the accuracy and effectiveness of the proposed method.