Interior Penalty Discontinuous Galerkin Research Articles

A symmetric interior-penalty discontinuous Galerkin isogeometric analysis spatial discretization of the self-adjoint angular flux form of the neutron transport equation

This paper presents the first application of a symmetric interior-penalty discontinuous Galerkin isogeometric analysis (SIP-DG-IGA) spatial discretization to the self-adjoint angular flux (SAAF) form of the multi-group neutron transport equation. The penalty parameters are determined, for general element types, from a mathematically rigorous coercivity analysis of the bilinear form. The proposed scheme produces a compact spatial discretization stencil. It also yields symmetric positive-definite (SPD) matrices, which can be efficiently solved using pre-conditioned conjugate gradient (PCG) solution algorithms. The proposed discretization scheme is verified using the method of manufactured solutions (MMS) and several nuclear reactor physics benchmark verification test cases. For sufficiently smooth elliptic problems, the proposed spatial discretization can exploit higher-order continuity, or k-refinement, of the NURBS basis to consistently yield greater numerical accuracy per degree of freedom (DoF) than standard h-refinement. Since this is a discontinuous scheme, it can also accurately model significant changes in the neutron scalar flux that may occur near the material interfaces of heterogeneous problems.

An optimization based limiter for enforcing positivity in a semi-implicit discontinuous Galerkin scheme for compressible Navier–Stokes equations

We consider an optimization-based limiter for enforcing positivity of internal energy in a semi-implicit scheme for solving gas dynamics equations. With Strang splitting, the compressible Navier–Stokes system is split into the compressible Euler equations, which are solved by the positivity-preserving Runge–Kutta discontinuous Galerkin (DG) method, and the parabolic subproblem, which is solved by Crank–Nicolson in time with interior penalty DG method. Such a scheme is at most second order accurate in time, high order accurate in space, conservative, and preserves positivity of density. To further enforce the positivity of internal energy, we impose an optimization-based limiter for the total energy variable to post-process DG polynomial cell averages. The optimization-based limiter can be efficiently implemented by the popular first order convex optimization algorithms such as the Douglas–Rachford splitting method by using nearly optimal algorithm parameters. Numerical tests suggest that the DG method with Qk basis and the optimization-based limiter is robust for demanding low-pressure problems such as high-speed flows.

Large eddy simulation based on an improved high-precision interior penalty discontinuous Galerkin method: flow past cylinders and airfoils

Large eddy simulation based on an improved high-precision interior penalty discontinuous Galerkin method: flow past cylinders and airfoils

A type of multigrid method for semilinear elliptic problems based on symmetric interior penalty discontinuous Galerkin method

AbstractThis article introduces a new kind of multigrid approach for semilinear elliptic problems, which is based on the symmetric interior penalty discontinuous Galerkin (SIPDG) method. We first give an optimal error estimate of the SIPDG method for the problem. Then, we design a type of multigrid method, which is called the multilevel correction method, and derive a priori error estimates. The primary idea of this method is to take the solution of the semilinear problem and utilize it to establish a sequence of solutions for associated linear boundary value problem on discontinuous finite element spaces and a newly defined low dimensional augmented subspace. Lastly, numerical experiments are offered to confirm the suggested method's precision and effectiveness.

A discontinuous Galerkin based multiscale method for heterogeneous elastic wave equations

In this paper, we develop a local multiscale model reduction strategy for the elastic wave equation in strongly heterogeneous media, which is achieved by solving the problem in a coarse mesh with multiscale basis functions. We use the interior penalty discontinuous Galerkin (IPDG) to couple the multiscale basis functions that contain important heterogeneous media information. The construction of efficient multiscale basis functions starts with extracting dominant modes of carefully defined spectral problems to represent important media feature, which is followed by solving a constraint energy minimization problem. Then a Petrov-Galerkin projection and systematization onto the coarse grid are applied. As a result, an explicit and energy-conserving scheme is obtained for fast online simulation. The method exhibits both coarse-mesh and spectral convergence as long as one appropriately chooses the oversampling size. We rigorously analyze the stability and convergence of the proposed method. Numerical results are provided to show the performance of the multiscale method and confirm the theoretical results.

Low regularity error analysis for an H(div)-conforming discontinuous Galerkin approximation of Stokes problem

In this paper, we derive an improved error estimate for the H(div)-conforming discontinuous Galerkin (DG) approximation of the Stokes equations, assuming only minimal regularity on the exact solution. The estimate relies on both a priori and a posteriori analysis, and thus is called a medius error analysis. More precisely, we proved an optimal order error estimate under the assumption (u,p)∈H1+s(Ω)×Hs(Ω) with any s∈(0,1]. Extension to the standard interior penalty DG methods is also explored. Finally, numerical results are provided to verify our theoretical findings.

A novel discontinuous Galerkin projection scheme for the hydrodynamics of nematic liquid crystals

This paper is focused on the numerical approximations for the hydrodynamic model of nematic liquid crystals. Under the framework of a splitting projection method, we propose a novel interior penalty discontinuous Galerkin (DG) method for solving the coupled system, which is employed by combining the scalar auxiliary variables (SAV) approach, implicit-explicit (IMEX) treatments and a rotational pressure-correction method. One prominent feature of the developed scheme here is by introducing an additional stabilization term artificially in liquid crystal equation to balance the explicit treatment for the coupling term, so that the computations of vector field from velocity field are decoupled. Hence a linear, fully decoupled and unconditionally energy-stable DG scheme can be achieved in a fully discrete manner. We show that the resulting scheme is uniquely solvable and unconditional energy stable, and the optimal error estimates of the proposed scheme are further proved theoretically. Finally, numerical results are carried out to demonstrate the accuracy and energy stability of our scheme.

Choice of Interior Penalty Coefficient for Interior Penalty Discontinuous Galerkin Method for Biot’s System By Employing Machine Learning

Choice of Interior Penalty Coefficient for Interior Penalty Discontinuous Galerkin Method for Biot’s System By Employing Machine Learning

A low-cost, penalty parameter-free, and pressure-robust enriched Galerkin method for the Stokes equations

This paper proposes a low-cost, penalty parameter-free, and pressure-robust Stokes solver based on the enriched Galerkin (EG) method with a discontinuous velocity enrichment function. The EG method employs the interior penalty discontinuous Galerkin (IPDG) formulation to weakly impose the continuity of the velocity function. However, despite its advantage of symmetry, the symmetric IPDG formulation requires a lot of computational effort to choose an optimal penalty parameter and compute different trace terms. To reduce such effort, we replace the derivatives of the velocity function with its weak derivatives computed by the geometric data of elements. Therefore, our modified EG (mEG) method is a penalty parameter-free numerical scheme that has reduced computational complexity and conserves the optimal convergence orders. Moreover, we achieve pressure robustness for the mEG method by employing a velocity reconstruction operator on the load vector on the right-hand side of the discrete system. The theoretical results are confirmed through numerical experiments with two- and three-dimensional examples.

Duality analysis of interior penalty discontinuous Galerkin methods under minimal regularity and application to the a priori and a posteriori error analysis of Helmholtz problems

We consider interior penalty discontinuous Galerkin discretizations of time-harmonic wave propagation problems modeled by the Helmholtz equation, and derive novel a priori and a posteriori estimates. Our analysis classically relies on duality arguments of Aubin–Nitsche type, and its originality is that it applies under minimal regularity assumptions. The estimates we obtain directly generalize known results for conforming discretizations, namely that the discrete solution is optimal in a suitable energy norm and that the error can be explicitly controlled by a posteriori estimators, provided the mesh is sufficiently fine.

Error analysis for discontinuous Galerkin method for time‐fractional Burgers' equation

The primary goal of this paper is to suggest a fully discrete numerical solution approach for the time‐fractional Burgers' equation. This paper will consider the fractional derivative in the Caputo sense. The time derivative of this equation will be discretized using the L2‐type discretization formula. The spatial variable is approximated by using the nonsymmetric interior penalty discontinuous Galerkin method. The proposed method is globally and unconditionally stable. The accuracy of the solution is evaluated using a convergence analysis. Computational experiments further confirm the accuracy and stability of the suggested strategy.

A numerical investigation on coupling of conforming and hybridizable interior penalty discontinuous Galerkin methods for fractured groundwater flow problems

The present paper focuses on the numerical modeling of groundwater flows in fractured porous media using the codimensional model description. Therefore, fractures are defined explicitly as a (d−1)-dimensional geometric object immersed in a d-dimensional region and can act arbitrarily as a drain or a barrier. We numerically investigate a novel numerical strategy combining distinctive classes of conforming and nonconforming high-order Galerkin methods, both eligible for static condensation. This procedure is here particularly relevant, leading to a smaller and sparser final system with coupled degrees of freedom solely on the mesh skeleton. Precisely, we combine an inspired hybridizable interior penalty discontinuous Galerkin (HIP) formulation inside the bulk region and a standard continuous Galerkin (CG) approximation on the fracture network. The distinctive discretization of corresponding PDEs and the coupling strategy are rigorously exposed, and the local and global matrix assemblies are detailed. Extensive numerical experiments are then achieved to prove the model’s performances for 2D/3D analytic and realistic benchmarks. Qualitative comparisons are also considered with other discretization methods and commercial software, such as comsol.

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.

A high order geometry conforming immersed finite element for elliptic interface problems

We present a high order immersed finite element (IFE) method for solving the elliptic interface problem with interface-independent meshes. The IFE functions developed here satisfy the interface conditions precisely and they have optimal approximation capabilities. The construction of this novel IFE space relies on a nonlinear transformation based on the Frenet–Serret frame of the interface to locally map it into a line segment, and this feature makes the process of constructing the IFE functions cost-effective and robust for any degree. This new class of immersed finite element functions is locally conforming with the usual weak form of the interface problem so that they can be employed in the standard interior penalty discontinuous Galerkin scheme without additional penalties on the interface. Numerical examples are provided to showcase the convergence properties of the method under h and p refinements.

A splitting discontinuous Galerkin projection method for the magneto-hydrodynamic equations

This paper is focused on the numerical approximations of the incompressible magneto-hydrodynamic equations. Our interest is to develop a novel decoupled, linear and unconditional energy stable fully-discrete scheme, which is achieved by the interior penalty discontinuous Galerkin (DG) method for spatial discretization, the stabilizing strategy and implicit-explicit (IMEX) scheme used to handle the nonlinear coupling terms, and a rotational pressure-correction method for the Navier-Stokes equations. We prove the unique solvability, unconditional energy stability and optimal error estimates of the proposed scheme rigorously. We further present several numerical examples to demonstrate the accuracy, stability, and efficiency of the proposed scheme numerically.

An interior penalty discontinuous Galerkin reduced order model for the variable coefficient advection–diffusion-reaction equation

An interior penalty discontinuous Galerkin reduced order model for the variable coefficient advection–diffusion-reaction equation

An hp-version interior penalty discontinuous Galerkin method for the quad-curl eigenvalue problem

An hp-version interior penalty discontinuous Galerkin method for the quad-curl eigenvalue problem

Efficient interior penalty discontinuous Galerkin projection method with unconditional energy stability and second-order temporal accuracy for the incompressible magneto-hydrodynamic system

In this paper, we propose a novel interior penalty discontinuous Galerkin projection method for the incompressible magneto-hydrodynamic equations. The scheme is employed by an implicit-explicit treatment of the nonlinear coupling terms and a second-order rotational pressure-correction scheme for dealing with the Navier-Stokes equations. One noteworthy aspect of this scheme is the introduction of an additional stabilization term to Maxwell's equations, which allows for the explicit treatment of the coupled nonlinear terms and the decoupling of computations for the magnetic and velocity fields, ultimately leading to the achievement of desired linearity, full decoupling, second-order accuracy in time, and unconditional energy stability. The obtained scheme is easy to implement because one only needs to solve a few decoupled linear equations at each time step. We rigorously prove the unique solvability and unconditional energy stability of the developed scheme and present a series of numerical examples to demonstrate the accuracy, stability, and efficiency of the proposed scheme.

A positivity-preserving implicit-explicit scheme with high order polynomial basis for compressible Navier–Stokes equations

In this paper, we are interested in constructing a scheme solving compressible Navier–Stokes equations, with desired properties including high order spatial accuracy, conservation, and positivity-preserving of density and internal energy under a standard hyperbolic type CFL constraint on the time step size, e.g., Δt=O(Δx). Strang splitting is used to approximate convection and diffusion operators separately. For the convection part, i.e., the compressible Euler equation, the high order accurate postivity-preserving Runge–Kutta discontinuous Galerkin method can be used. For the diffusion part, the equation of internal energy instead of the total energy is considered, and a first order semi-implicit time discretization is used for the ease of achieving positivity. A suitable interior penalty discontinuous Galerkin method for the stress tensor can ensure the conservation of momentum and total energy for any high order polynomial basis. In particular, positivity can be proven with Δt=O(Δx) if the Laplacian operator of internal energy is approximated by the Qk spectral element method with k=1,2,3. So the full scheme with Qk (k=1,2,3) basis is conservative and positivity-preserving with Δt=O(Δx), which is robust for demanding problems such as solutions with low density and low pressure induced by high-speed shock diffraction. Even though the full scheme is only first order accurate in time, numerical tests indicate that higher order polynomial basis produces much better numerical solutions, e.g., better resolution for capturing the roll-ups during shock reflection.

Dispersion reduction in Feng and Wu’s IPDG method

We give an analytical justification of the determination of an important parameter involved in Feng and Wu’s solution of the Helmholtz equation by an Interior Penalty Discontinuous Galerkin (IPDG) method (Feng and Wu, 2009). This parameter was determined in this reference from a large number of numerical tests for a mesh composed of equal equilateral triangles at a fixed frequency. It plays an essential role in the reduction the dispersion error for a polynomial approximation of degree 1. We show how this justification makes it possible to slightly improve it and to handle other structured meshes. The analytical results are evaluated by several numerical experiments, showing in particular that the parameter determined in principle for equal equilateral triangles also eliminates dispersion for unstructured, well-smoothed meshes.