Interior Penalty Discontinuous Galerkin Research Articles

Adaptivity and Blow-Up Detection for Nonlinear Evolution Problems

This work is concerned with the development of a space-time adaptive numerical method, based on a rigorous a posteriori error bound, for a semilinear convection-diffusion problem which may exhibit blow-up in finite time. More specifically, a posteriori error bounds are derived in the $L^{\infty}(L^2)+L^2(H^1)$-type norm for a first order in time implicit-explicit interior penalty discontinuous Galerkin in space discretization of the problem, although the theory presented is directly applicable to the case of conforming finite element approximations in space. The choice of the discretization in time is made based on a careful analysis of adaptive time-stepping methods for ODEs that exhibit finite time blow-up. The new adaptive algorithm is shown to accurately estimate the blow-up time of a number of problems, including one which exhibits regional blow-up.

Strong convergence of discrete DG solutions of the heat equation

AbstractA convergence analysis to the weak solution is derived for interior penalty discontinuous Galerkin methods applied to the heat equation in two and three dimensions under general mixed boundary conditions. Strong convergence is established in the DG norm, as well as in the

$hp$-Adaptation Driven by Polynomial-Degree-Robust A Posteriori Error Estimates for Elliptic Problems

We devise and study experimentally adaptive strategies driven by a posteriori error estimates to select automatically both the space mesh and the polynomial degree in the numerical approximation of diffusion equations in two space dimensions. The adaptation is based on equilibrated flux estimates. These estimates are presented here for inhomogeneous Dirichlet and Neumann boundary conditions, for spatially-varying polynomial degree, and for mixed rectangular-triangular grids possibly containing hanging nodes. They deliver a global error upper bound with constant one and, up to data oscillation, error lower bounds on element patches with a generic constant only dependent on the mesh regularity and with a computable bound. We numerically asses the estimates and several hp-adaptive strategies using the interior penalty discontinuous Galerkin method. Asymptotic exactness is observed for all the symmetric, nonsymmetric (odd degrees), and incomplete variants on non-nested unstructured triangular grids for a smooth solution and uniform refinement. Exponential convergence rates are reported on nonmatching triangular grids for the incomplete version on several benchmarks with a singular solution and adaptive refinement.

A High-Order Symmetric Interior Penalty Discontinuous Galerkin Scheme to Simulate Vortex Dominated Incompressible Fluid Flow

A high-order Symmetric Interior Penalty discontinuous Galerkin (SIPG) method has been used for solving the incompressible Navier-Stokes equation. We apply the temporal splitting scheme in time and the SIPG discretization in space with the local Lax-Friedrichs flux for the discretization of nonlinear terms. A fully discrete semi-implicit splitting scheme has been presented and high-order discontinuous Galerkin (DG) finite elements are available. Under a constraint of the CFL condition, two benchmark problems in 2D are investigated: one is a lid-driven cavity flow to verify the high-order discontinuous Galerkin method is accurate and robust; the other is a flow past a circular cylinder, for which we mainly check the Strouhal numbers with the von Karman vortex street, and also simulate the boundary layers with walls and corresponding dynamical behavior with Neumann conditions on the top and bottom boundaries, respectively. We predict the Strouhal number for the range of Reynolds number 50 ≤ Re ≤ 400, making a comparison between the predicted values by our numerical method and the referenced values from physical experiments.

Discontinuous Galerkin immersed finite element methods for parabolic interface problems

In this article, interior penalty discontinuous Galerkin methods using immersed finite element functions are employed to solve parabolic interface problems. Typical semi-discrete and fully discrete schemes are presented and analyzed. Optimal convergence for both semi-discrete and fully discrete schemes is proved. Some numerical experiments are provided to validate our theoretical results.

$$C^0$$ C 0 IP Methods for the Transmission Eigenvalue Problem

We consider a non-selfadjoint fourth order eigenvalue problem using a discontinuous Galerkin (DG) method. For high order problems, DG methods are competitive since they use simple basis functions and have less degrees of freedom. The numerical implementation is much easier compared with classical conforming finite element methods. In this paper, we propose an interior penalty discontinuous Galerkin method using $$C^0$$C0 Lagrange elements ($$C^0$$C0IP) for the transmission eigenvalue problem and prove the optimal convergence. The method is applied to several examples and its effectiveness is validated.

Structure preserving integration and model order reduction of skew-gradient reaction–diffusion systems

Activator-inhibitor FitzHugh-Nagumo (FHN) equation is an example for reaction-diffusion equations with skew-gradient structure. We discretize the FHN equation using symmetric interior penalty discontinuous Galerkin (SIPG) method in space and average vector field (AVF) method in time. The AVF method is a geometric integrator, i.e. it preserves the energy of the Hamiltonian systems and energy dissipation of the gradient systems. In this work, we show that the fully discrete energy of the FHN equation satisfies the mini-maximizer property of the continuous energy for the skew-gradient systems. We present numerical results with traveling fronts and pulses for one dimensional, two coupled FHN equations and three coupled FHN equations with one activator and two inhibitors in skew-gradient form. Turing patterns are computed for fully discretized two dimensional FHN equation in the form of spots and labyrinths. Because the computation of the Turing patterns is time consuming for different parameters, we applied model order reduction with the proper orthogonal decomposition (POD). The nonlinear term in the reduced equations is computed using the discrete empirical interpolation (DEIM) with SIPG discretization. Due to the local nature of the discontinuous Galerkin (DG) method, the nonlinear terms can be computed more efficiently than for the continuous finite elements. The reduced solutions are very close to the fully discretized ones. The efficiency and accuracy of the POD and POD-DEIM reduced solutions are shown for the labyrinth-like patterns.

Interior Penalty Discontinuous Galerkin Based Isogeometric Analysis for Allen-Cahn Equations on Surfaces

AbstractWe propose a method that combines Isogeometric Analysis (IGA) with the interior penalty discontinuous Galerkin (IPDG) method for solving the Allen-Cahn equation, arising from phase transition in materials science, on three-dimensional (3D) surfaces consisting of multiple patches. DG ideology is adopted at patch level, i.e., we employ the standard IGA within each patch, and employ the IPDG method across the patch interfaces. IGA is very suitable for solving Partial Differential Equations (PDEs) on (3D) surfaces and the IPDG method is used to glue the multiple patches together to get the right solution. Our method takes advantage of both IGA and the IPDG method, which allows us to design a superior semi-discrete (in time) IPDG scheme. First and most importantly, the time-consuming mesh generation process in traditional Finite Element Analysis (FEA) is no longer necessary and refinements, includingh-refinement andp-refinement which both maintain the original geometry, can be easily performed at any level. Moreover, the flexibility of the IPDG method makes our method very easy to handle cases with non-conforming patches and different degrees across the patch interfaces. Additionally, the geometrical error is eliminated (for all conic sections) or significantly reduced at the beginning due to the geometric flexibility of IGA basis functions, especially the use of multiple patches. Finally, this method can be easily formulated and implemented. We present our semi-discrete IPDG scheme after generally describe the problem, and then briefly introduce the time marching method employed in this paper. Theoretical analysis is carried out to show that our method satisfies a discrete energy law, and achieves the optimal convergence rate with respect to theL2norm. Furthermore, we propose an elliptic projection operator on (3D) surfaces and prove an approximation error estimate which are vital for us to obtain the error estimate in theL2norm. Numerical tests are given to validate the theory and gauge the good performance of our method.

Non-conforming Galerkin finite element methods for local absorbing boundary conditions of higher order

A new non-conforming finite element discretization methodology for second order elliptic partial differential equations involving higher order local absorbing boundary conditions in 2D and 3D is proposed. The novelty of the approach lies in the application of C0-continuous finite element spaces, which is the standard discretization of second order operators, to the discretization of boundary differential operators of order four and higher. For each of these boundary operators, additional terms appear on the boundary nodes in 2D and on the boundary edges in 3D, similarly to interior penalty discontinuous Galerkin methods, which leads to a stable and consistent formulation. In this way, no auxiliary variables on the boundary have to be introduced and trial and test functions of higher smoothness along the boundary are not required. As a consequence, the method leads to lower computational costs for discretizations with higher order elements and is easily integrated in high-order finite element libraries. A priorih-convergence error estimates show that the method does not reduce the order of convergence compared to usual Dirichlet, Neumann or Robin boundary conditions if the polynomial degree on the boundary is increased simultaneously. A series of numerical experiments illustrates the utility of the method and validates the theoretical convergence results.

Superconvergence property of an over-penalized discontinuous Galerkin finite element gradient recovery method

A polynomial preserving recovery method is introduced for over-penalized symmetric interior penalty discontinuous Galerkin solutions to a quasi-linear elliptic problem. As a post-processing method, the polynomial preserving recovery is superconvergent for the linear and quadratic elements under specified meshes in the regular and chevron patterns, as well as general meshes satisfying Condition(ϵ,σ). By means of the averaging technique, we prove the polynomial preserving recovery method for averaged solutions is superconvergent, satisfying similar estimates as those for conforming finite element methods. We deduce superconvergence of the recovered gradient directly from discontinuous solutions and naturally construct an a posteriori error estimator. Consequently, the a posteriori error estimator based on the recovered gradient is asymptotically exact. Extensive numerical results consistent with our analysis are presented.

Preasymptotic Error Analysis of High Order Interior Penalty Discontinuous Galerkin Methods for the Helmholtz Equation with High Wave Number

A preasymptotic error analysis of the interior penalty discontinuous Galerkin (IPDG) method of high order for Helmholtz equation with the first order absorbing boundary condition in two and three dimensions is proposed. We derive the $$H^1$$H1- and $$L^2$$L2- error estimates with explicit dependence on the wave number k. In particular, it is shown that if $$k(kh)^{2p}$$k(kh)2p is sufficiently small, then the pollution errors of IPDG method in $$H^1$$H1-norm are bounded by $$O(k(kh)^{2p})$$O(k(kh)2p), which coincides with the phase error of the finite element method obtained by existent dispersion analyses on Cartesian grids, where h is the mesh size, p is the order of the approximation space and is fixed. Numerical tests are provided to verify the theoretical findings and to illustrate great capability of the symmetric IPDG method in reducing the pollution effect.

Energy norm error estimates for averaged discontinuous Galerkin methods: Multidimensional case

A mathematical analysis is presented for a class of interior penalty (IP) discontinuous Galerkin approximations of elliptic boundary value problems. In the framework of the present theory one can derive some overpenalized IP bilinear forms avoiding the notion of fluxes and artificial penalty terms. The main idea is to start from bilinear forms for the local average of discontinuous approximations which are rewritten using the theory of distributions. It is pointed out that a class of overpenalized IP bilinear forms can be obtained using a perturbation of these. Also, error estimations can be derived between the local averages of the discontinuous approximations and the analytic solution in the H1-seminorm. Using the local averages, the analysis is performed in a conforming framework without any assumption on extra smoothness for the solution of the original boundary value problem.

Convergence of a high order method in time and space for the miscible displacement equations

A numerical method is formulated and analyzed for solving the miscible displacement problem under low regularity assumptions. The scheme employs discontinuous Galerkin time stepping with mixed and interior penalty discontinuous Galerkin finite elements in space. The numerical approx- imations of the pressure, velocity, and concentration converge to the weak solution as the mesh size and time step tend to zero. To pass to the limit a compactness theorem is developed which generalizes the Aubin−Lions theorem to accommodate discontinuous functions both in space and in time.

Numerical solution of steady-state groundwater flow and solute transport problems: Discontinuous Galerkin based methods compared to the Streamline Diffusion approach

In this study, we consider the simulation of subsurface flow and solute transport processes in the stationary limit. In the convection-dominant case, the numerical solution of the transport problem may exhibit non-physical diffusion and under- and overshoots. For an interior penalty discontinuous Galerkin (DG) discretization, we present a $h$-adaptive refinement strategy and, alternatively, a new efficient approach for reducing numerical under- and overshoots using a diffusive $L^2$-projection. Furthermore, we illustrate an efficient way of solving the linear system arising from the DG discretization. In $2$-D and $3$-D examples, we compare the DG-based methods to the streamline diffusion approach with respect to computing time and their ability to resolve steep fronts.

Pre-asymptotic error analysis of [formula omitted]-interior penalty discontinuous Galerkin methods for the Helmholtz equation with large wave number

In this paper we shall consider to improve the pre-asymptotic stability and error estimates of some hp-interior penalty discontinuous Galerkin (hp-IPDG) methods for the Helmholtz equation with the first order absorbing boundary condition in two and three dimensions given in Feng and Wu (2011). The proposed hp-IPDG methods are defined using a sesquilinear form which is not only mesh-dependent (or h-dependent) but also degree-dependent (or p-dependent). By using a modified duality argument given in Zhu and Wu (2013), pre-asymptotic error estimates are improved for the proposed hp-IPDG methods under the condition of khp≤C0(pk)1p+1 in this paper, where C0 is some constant independent of k,h,p, and the penalty parameters. It is shown that the pollution error of the method in the broken H1-norm is O(k2p+1h2p) if p=O(1) which coincides with existent dispersion analyses for the DG method on Cartesian grids. Numerical tests are provided to verify the theoretical findings and to illustrate great capability of the IPDG method in reducing the pollution effect.

Comparison Results of NonstandardP2Finite Element Methods for the Biharmonic Problem

As modern variant of nonconforming schemes, discontinuous Galerkin finite element meth- ods appear to be highly attractive for fourth-order elliptic PDEs. There exist various modifications and the most prominent versions with first-order convergence properties are the symmetric interior penalty DG method and the C 0 interior penalty method which may compete with the classical Morley nonconforming FEM on triangles. Those schemes differ in their various jump and penalisation terms and also in the norms. This paper proves that the best-approximation errors of all the three schemes are equivalent in the sense that their minimal error in the respective norm and the optimal choice of a discrete approximation can be bounded from below and above by each other. The equivalence constants do only depend on the minimal angle of the triangulation and the penalisation parameter of the schemes; they are independent of any regularity requirement and hold for an arbitrarily coarse mesh.

Discontinuous Galerkin Finite Element Methods for Numerical Simulations of Thermoelasticity

In this article, the discontinuous Galerkin (DG) finite element methods are extended to numerical simulations of uncoupling thermoelasticity. A thermoelastic formulation of interior penalty DG (IP-DG) method is presented and aspects of the numerical implementation are discussed in matrix form. The feasibility of the method for general thermoelastic simulations is validated through some typical test cases. Investigations are performed in which the method is applied to problems with thermal stress discontinuities induced by non-homogeneous inclusions, illustrating that reasonably ranged magnitude of IP term can well deal with the problems, and IP parameter increasing can effectively suppress the non-physical oscillations around discontinuities although its excess may affect solution accuracy in smooth region to some extent.

Fully discrete IPDG–HMM for multiscale Richards equation of unsaturated flow in porous media

In this paper, we propose a fully discrete method for the multiscale Richards equation which describes the flow transport in unsaturated heterogeneous porous media. Under the framework of heterogeneous multiscale method (HMM), a fully discrete interior penalty discontinuous Galerkin finite element method (IPDG–FEM) is applied over a macro-scale mesh. The fully discrete method means that it takes into account not only the fully macro-scale discretization but also the fully micro-scale discretization for local cell problems. Error estimates between the numerical solution and the solution of homogenized problem are derived under the assumption that the permeability is periodic. Numerical experiments with periodic and random permeability are carried out both for Gardener model and van Genuchten–Mualem model of Richards equation to show the efficiency and accuracy of the proposed method.

Superconvergence Analysis of a Full-Discrete Combined Mixed Finite Element and Discontinuous Galerkin Approximation for an Incompressible Miscible Displacement Problem

For an incompressible miscible displacement problem, an effective time-stepping procedure is proposed. The mixed finite element method is applied to the flow equation with uniform meshes, and the transport equation is solved by a full discretized interior penalty discontinuous Galerkin method with regular partitions. Convolution of the Darcy velocity approximation with the Bramble-Schatz kernel function and averaging are applied in the evaluation of the coefficients in the Galerkin procedure for the concentration. A superconvergence estimate is presented.

A massively parallel nonoverlapping additive Schwarz method for discontinuous Galerkin discretization of elliptic problems.

A second order elliptic problem with discontinuous coefficient in 2-D or 3-D is considered. The problem is discretized by a symmetric weighted interior penalty discontinuous Galerkin finite element method with nonmatching simplicial elements and piecewise linear functions. The resulting discrete problem is solved by a two-level additive Schwarz method with a relatively coarse grid and with local solves restricted to subdomains which can be as small as single element. In this way the method has a potential for a very high level of fine grained parallelism. Condition number estimate depending on the relative sizes of the underlying grids is provided. The rate of convergence of the method is independent of the jumps of the coefficient if its variation is moderate inside coarse grid substructures or on local solvers’ subdomain boundaries. Numerical experiments are reported which confirm theoretical results.