Symmetric Interior Penalty Discontinuous Galerkin Research Articles

A Phase-Space Discontinuous Galerkin Scheme for the Radiative Transfer Equation in Slab Geometry

Abstract We derive and analyze a symmetric interior penalty discontinuous Galerkin scheme for the approximation of the second-order form of the radiative transfer equation in slab geometry. Using appropriate trace lemmas, the analysis can be carried out as for more standard elliptic problems. Supporting examples show the accuracy and stability of the method also numerically, for different polynomial degrees. For discretization, we employ quad-tree grids, which allow for local refinement in phase-space, and we show exemplary that adaptive methods can efficiently approximate discontinuous solutions. We investigate the behavior of hierarchical error estimators and error estimators based on local averaging.

High-order Accurate Beam Models Based on Discontinuous Galerkin Methods

A novel high-order accurate approach to the analysis of beam structures with bulk and thin-walled cross-sections is presented. The approach is based on the use of a variable-order polynomial expansion of the displacement field throughout both the beam cross-section and the length of the beam elements. The corresponding weak formulation is derived using the symmetric Interior Penalty discontinuous Galerkin method, whereby the continuity of the solution at the interface between contiguous elements as well as the application of the boundary conditions is weakly enforced by suitably defined boundary terms. The accuracy and the flexibility of the proposed approach are assessed by modeling slender and short beams with standard square cross-sections and airfoil-shaped thin-walled cross-sections subjected to bending, torsional and aerodynamic loads. The comparison between the obtained numerical results and those available in the literature or computed using a standard finite-element method shows that the present method allows recovering three-dimensional distributions of displacement and stress fields using a significantly reduced number of degrees of freedom.

Reduced Order Modelling of Shigesada-Kawasaki-Teramoto Cross-Diffusion Systems

Shigesada-Kawasaki-Teramoto (SKT) is the most known equation in population ecology for nonlinear cross-diffusion systems. The full order model (FOM) of the SKT system is constructed using symmetric interior penalty discontinuous Galerkin method (SIPG) in space and the semi-implicit Euler method in time. The reduced order models (ROMs) are solved using proper orthogonal decomposition (POD) Galerkin projection. Discrete empirical interpolation method (DEIM) is used to solve the nonlinearities of the SKT system. Numerical simulations show the accuracy and efficiency of the POD and POD-DEIM reduced solutions for the SKT system.

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.

A discontinuous Galerkin method for sequences of earthquakes and aseismic slip on multiple faults using unstructured curvilinear grids

SUMMARYPhysics-based simulations provide a path to overcome the lack of observational data hampering a holistic understanding of earthquake faulting and crustal deformation across the vastly varying space–time scales governing the seismic cycle. However, simulations of sequences of earthquakes and aseismic slip (SEAS) including the complex geometries and heterogeneities of the subsurface are challenging. We present a symmetric interior penalty discontinuous Galerkin (SIPG) method to perform SEAS simulations accounting for the aforementioned challenges. Due to the discontinuous nature of the approximation, the spatial discretization natively provides a means to impose boundary and interface conditions. The method accommodates 2-D and 3-D domains, is of arbitrary order, handles subelement variations in material properties and supports isoparametric elements, that is, high-order representations of the exterior boundaries, interior material interfaces and embedded faults. We provide an open-source reference implementation, Tandem, that utilizes highly efficient kernels for evaluating the SIPG linear and bilinear forms, is inherently parallel and well suited to perform high-resolution simulations on large-scale distributed memory architectures. Additional flexibility and efficiency is provided by optionally defining the displacement evaluation via a discrete Green’s function approach, exploiting advantages of both the boundary integral and volumetric methods. The optional discrete Green’s functions are evaluated once in a pre-computation stage using algorithmically optimal and scalable sparse parallel solvers and pre-conditioners. We illustrate the characteristics of the SIPG formulation via an extensive suite of verification problems (analytic, manufactured and code comparison) for elastostatic and quasi-dynamic problems. Our verification suite demonstrates that high-order convergence of the discrete solution can be achieved in space and time and highlights the benefits of using a high-order representation of the displacement, material properties and geometries. We apply Tandem to realistic demonstration models consisting of a 2-D SEAS multifault scenario on a shallowly dipping normal fault with four curved splay faults, and a 3-D intersecting multifault scenario of elastostatic instantaneous displacement of the 2019 Ridgecrest, CA, earthquake sequence. We exploit the curvilinear geometry representation in both application examples and elucidate the importance of accurate stress (or displacement gradient) representation on-fault. This study entails several methodological novelties. We derive a sharp bound on the smallest value of the SIPG penalty ensuring stability for isotropic, elastic materials; define a new flux to incorporate embedded faults in a standard SIPG scheme; employ a hybrid multilevel pre-conditioner for the discrete elasticity problem; and demonstrate that curvilinear elements are specifically beneficial for volumetric SEAS simulations. We show that our method can be applied for solving interesting geophysical problems using massively parallel computing. Finally, this is the first time a discontinuous Galerkin method is published for the numerical simulations of SEAS, opening new avenues to pursue extreme scale 3-D SEAS simulations in the future.

Strong optimal error estimates of discontinuous Galerkin method for multiplicative noise driving nonlinear SPDEs

AbstractThis paper investigates the strong convergence of a fully discrete numerical method for the stochastic partial differential equations driven by multiplicative noise. The fully discrete space–time approximation consists of the symmetric interior penalty discontinuous Galerkin method for the spatial discretization and the implicit Euler method for the temporal discretization. Rather than the usual semi group analysis techniques, in this paper, we present an analysis framework in the variational formulation by introducing new weak variational approximation techniques. Some error estimates in a strong sense are established for the proposed fully discrete scheme. The optimal convergence rates are then obtained in both space and time. Numerical results for the nonlinear stochastic partial differential equations are finally presented to confirm the theoretical results.

A new symmetric interior penalty discontinuous Galerkin formulation for the Serre–Green–Naghdi equations

AbstractIn this work, we investigate the construction of a new discontinuous Galerkin discrete formulation to approximate the solution of Serre–Green–Naghdi (SGN) equations in the one‐dimensional horizontal framework. Such equations describe the time evolution of shallow water free surface flows in the fully nonlinear and weakly dispersive asymptotic approximation regime. A new non‐conforming discrete formulation belonging to the family of symmetric interior penalty discontinuous Galerkin methods is introduced to accurately approximate the solutions of the second order elliptic operator occurring in the SGN equations. We show that the corresponding discrete bilinear form enjoys some consistency and coercivity properties, thus ensuring that the corresponding discrete problem is well‐posed. The resulting global discrete formulation is then validated through an extended set of benchmarks, including convergence studies and comparisons with data taken from experiments.

Pointwisea posteriorierror analysis of a discontinuous Galerkin method for the elliptic obstacle problem

AbstractWe present a posteriori error analysis in the supremum norm for the symmetric interior penalty discontinuous Galerkin method for the elliptic obstacle problem. We construct discrete barrier functions based on appropriate corrections of the conforming part of the solution obtained via a constrained averaging operator. The corrector function accounts properly for the nonconformity of the approximation and it is estimated by direct use of the Green’s function of the unconstrained elliptic problem. The use of the continuous maximum principle guarantees the validity of the analysis without mesh restrictions but with shape regularity. The proposed residual-type estimators are shown to be reliable and efficient. Numerical results in two dimensions are included to verify the theory and validate the performance of the error estimator.

Equal Higher Order Analysis of an Unfitted Discontinuous Galerkin Method for Stokes Flow Systems

In this work, we analyze an unfitted discontinuous Galerkin discretization for the numerical solution of the Stokes system based on equal higher-order discontinuous velocities and pressures. This approach combines the best from both worlds, firstly the advantages of a piece-wise discontinuous high–order accurate approximation and secondly the advantages of an unfitted to the true geometry grid around possibly complex objects and/or geometrical deformations. Utilizing a fictitious domain framework, the physical domain of interest is embedded in an unfitted background mesh and the geometrically unfitted discretization is built upon symmetric interior penalty discontinuous Galerkin formulation. To enhance stability we enrich the discrete variational formulation with a pressure stabilization term. Moreover, the present contribution adopts high order ghost penalty strategies to address the ill conditioning of the system matrix caused by small truncated elements with respect to the unfitted boundary. Motivated by continuous unfitted FEM (Burman and Hansbo in ESAIM Math Model Numer Anal 48(3):859–874, 2014; Massing et al. in J Sci Comput 61:604–628, 2014; Massing et al. in Numer Math 128:73–101, 2014) along with other unfitted mesh surveys grounded on discontinuous spaces (Becker et al. in Comput Methods Appl Mech Eng 198(41–44):3352–3360, 2009; Gürkan and Massing in Comput Methods Appl Mech Eng 348:466–499, 2019; Gürkan et al. in SIAM J Sci Comput 42(5):A2620–A2654, 2020; Massing in A cut discontinuous Galerkin method for coupled bulk-surface problems, Chapter in UCLWorkshop volume on "Geometrically Unfitted Finite Element Methods", Lecture Notes in Computational Science and Engineering, Springer, Cham, pp 259–279, 2017), we use proper velocity and pressure ghost penalties defined on faces of cut cells to establish a robust high-order method, in spite of the cell agglomeration technique usually applied on dG methods. The current presentation should prove valuable in engineering applications where special emphasis is placed on the optimal effective approximation attaining much smaller relative errors in coarser meshes. Inf-sup stability, the optimal order of convergence, and the condition number sensitivity with respect to cut configuration are investigated. Numerical examples verify the theoretical results.

IETI-DP methods for discontinuous Galerkin multi-patch Isogeometric Analysis with T-junctions

We study Dual–Primal Isogeometric Tearing and Interconnecting (IETI-DP) solvers for non-conforming multi-patch discretizations of a generalized Poisson problem. We realize the coupling between the patches using a symmetric interior penalty discontinuous Galerkin (SIPG) approach. Previously, we have assumed that the interfaces between patches always consist of whole edges. In this paper, we drop this requirement and allow T-junctions. This extension is vital for the consideration of sliding interfaces, for example between the rotor and the stator of an electrical motor. One critical part for the handling of T-junctions in IETI-DP solvers is the choice of the primal degrees of freedom. We propose to add all basis functions that are non-zero at any of the vertices to the primal space. Since there are several such basis functions at any T-junction, we call this concept “fat vertices”. For this choice, we show a condition number bound that coincides with the bound for the conforming case.

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>

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.

High order discontinuous Galerkin method for reduced flow models in fractured porous media

We construct a new symmetric interior penalty discontinuous Galerkin (SIPDG)-method for incompressible flow in fractured porous media. The method is developed for computing accurate approximations of the reduced flow model, where fractures are treated as lower dimensional objects. Unlike previous formulations, the methodology proposed herein shows ability to capture the asymptotic limits of highly permeable and fracture seals, also covering a wider range of values of the quadrature integration parameter which appears in the pressure jumps across the fractures. As a first novel contribution we should mention that the method was developed for specific interface condition in the reduced problem for which known in literature DG method are not applicable. As a second novelty we propose a new penalty technique for stabilization of the method and obtain explicit estimate for the penalty parameters associated with flow in matrix and fractures in order to achieve stability. And finally we derive new high order hp type a priori error estimates for the numerical solution in the energy norm. Numerical results illustrate the performance of the proposed SIPDG-method in simulating discrete fracture models.

A robust discontinuous Galerkin scheme on anisotropic meshes

Discontinuous Galerkin (DG) methods are extensions of the usual Galerkin finite element methods. Although there are vast amount of studies on DG methods, most of them have assumed shape-regularity conditions on meshes for both theoretical error analysis and practical computations. In this paper, we present a new symmetric interior penalty DG scheme with a modified penalty term. We show that, without imposing the shape-regularity condition on the meshes, the new DG scheme inherits all of the good properties of standard DG methods, and is thus robust on anisotropic meshes. Numerical experiments confirm the theoretical error estimates obtained.

Discontinuous Galerkin solutions for elliptic problems on polygonal grids using arbitrary-order Bernstein-Bézier functions

In this paper, we present a symmetric interior penalty discontinuous Galerkin finite element discretization for the numerical solution of second-order elliptic partial differential equations on general polygonal meshes using a reduced-space polygonal Bernstein-Bézier functions. They form a space of interpolatory functions (vice local polynomial functions), which satisfy the Lagrange property at their interpolation points. This can enable the use of efficient DGFEM physics-based diffusion preconditioners for the first-order linear transport equation on arbitrary polygons in the computationally-challenging asymptotic diffusion limit. On a polygonal element with n vertices and polynomial degree p, there are n vertex functions, (p−1) functions per edge, and (p−1)(p−2)/2 interior functions that span the {xayb}(a+b)≤p space of bivariate monomials. Numerical experiments highlighting the performance of the proposed method are presented through uniform and adaptive h, p, and hp refinement strategies. Appropriate error rates are observed including hp adaptivity yielding exponential convergence in the presence of singularities in the solution.

The error analysis of linearized discontinuous Galerkin finite element method for incompressible miscible displacement in porous media

We present a linearization scheme for discontinuous Galerkin method for solving the fully coupled one-phase flow problem. The pressure and concentration equations are solved by symmetric interior penalty Discontinuous Galerkin (SIPG) method and we discretized the model using (DGM) space and backward Euler in time. We prove the error estimate in the energy norm.

Slope limiting the velocity field in a discontinuous Galerkin divergence-free two-phase flow solver

Solving the Navier–Stokes equations when the density field contains a large and sharp discontinuity—such as a water/air free surface—is numerically challenging. Convective instabilities cause Gibbs oscillations which quickly destroy the solution. We investigate the use of slope limiters for the velocity field to overcome this problem in a way that does not compromise on the mass-conservation properties. The equations are discretised using a symmetric interior-penalty discontinuous Galerkin finite element method that is divergence-free to machine precision.A slope limiter made specifically for exactly divergence-free (solenoidal) fields is presented and used to illustrate the difficulties in obtaining convectively stable fields that are also exactly solenoidal. The lessons learned from this are applied in constructing a simpler method based on the use of an existing scalar slope limiter applied to each velocity component.We show by numerical examples how both presented slope limiting methods are vastly superior to the naive non-limited method. The methods can solve difficult two-phase problems with high density-ratios and high Reynolds numbers—typical for marine and offshore water/air simulations—in a way that conserves mass and stops unbounded energy growth caused by the Gibbs phenomenon.

Calculating basin-scale free oscillations in lakes on a rotating Earth

A novel method is presented for numerically calculating the free modes of oscillation in closed rotating water basins of arbitrary shape based on a symmetric interior penalty discontinuous Galerkin (SIP-DG) finite element method (FEM). The new method is flexible and is shown to work well for any basin that has been appropriately triangulated with finite element mesh generation software. To illustrate the method's capabilities, the modes of both idealized and real-world lakes are computed and discussed.

Weak discrete maximum principle and $$L^\infty $$ analysis of the DG method for the Poisson equation on a polygonal domain

We derive several $L^\infty$ error estimates for the symmetric interior penalty (SIP) discontinuous Galerkin (DG) method applied to the Poisson equation in a two-dimensional polygonal domain. Both local and global estimates are examined. The weak maximum principle (WMP) for the discrete harmonic function is also established. We prove our $L^\infty$ estimates using this WMP and several $W^{2,p}$ and $W^{1,1}$ estimates for the Poisson equation. Numerical examples to validate our results are also presented.

A parabolic level set reinitialisation method using a discontinuous Galerkin discretisation

Level set reinitialisation is a part of the level set methodology which allows one to generate, at any point during level set evolution, a level set function which is a signed distance function to its own zero isocontour. Whilst not in general a required condition, maintaining the level set function as a signed distance function is often desirable as it removes a known source of numerical instability. This paper presents a novel level set reinitialisation method based on the solution of a nonlinear parabolic PDE. The PDE is discretised using a symmetric interior penalty discontinuous Galerkin method in space, and an implicit Euler method in time. Also explored are explicit and semi-implicit time discretisations, however, numerical experiments demonstrate that such methods suffer from severe time step restrictions, leading to prohibitively large numbers of iterations required to achieve convergence. The proposed method is shown to be high-order accurate through a number of numerical examples. More specifically, the presented experimental orders of convergence align with the well established optimal convergence rates for the symmetric interior penalty method; that is the error in the L2 norm decreases proportionally to hp+1 and the error in the DG norm decreases proportionally to hp.