Interior Penalty Discontinuous Galerkin Method Research Articles

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.

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

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.

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.

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 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 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

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.

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.

Interior penalty discontinuous Galerkin methods for the velocity-pressure formulation of the Stokes spectral problem

Interior penalty discontinuous Galerkin methods for the velocity-pressure formulation of the Stokes spectral problem

High-order enforcement of jumps conditions between compressible viscous phases: An extended interior penalty discontinuous Galerkin method for sharp interface simulation

In this paper, we develop a high-order method for the steady two-phase compressible Navier–Stokes equations closed by generic equations of state, adapted to gas–liquid flows. Our framework relies on the extended discontinuous Galerkin method that is tailored here to a general viscous compressible two-phase configuration. While the imposition of interface jumps conditions is commonly recognized as one of the main difficulties to overcome in the context of geometrically unfitted methods, the setting considered in this work makes this aspect even more challenging. To enforce the non-linear coupling conditions at the interface in a sharp manner, we devise a novel strategy combining multiphase Riemann solvers to handle the convective fluxes across the interface and a weighted stabilized Nitsche’s method for the viscous jumps. The weak enforcement of the jumps conditions along with implicit steady-state iterations and cell-agglomeration procedure make the approach robust to treat arbitrary large contrast phase interface problems governed by stiff models, irrespective of the interface location on the mesh. The study is here restricted to static interfaces. Based on the method of manufactured solution, reference solutions are designed. This allows to conduct a detailed numerical study investigating the influence of several numerical parameters. It is shown that to reach optimal error convergence, the use of pressure-velocity-temperature variables set, appropriate all-speed bulk Riemann solver and Symmetric Interior Penalty method combined with tensorial penalty is necessary.

A posteriori error estimator for mixed interior penalty discontinuous Galerkin finite element method for the [formula omitted](curl)-elliptic problems

In this paper, we design the first residual type a posteriori error estimator for mixed interior penalty discontinuous Galerkin method for the H(curl)-elliptic problems. Then we prove that our indicator is both reliable and efficient. At last, we present some numerical experiments to validate the performance of the indicator within an adaptive mesh refinement procedure.

Turbulence simulations with an improved interior penalty discontinuous Galerkin method and SST k-ω model

The feasibility of high-precision interior penalty discontinuous Galerkin (IPDG) method is still questionable when SST k-ω model and its compressibility correction are adopted. To this end, turbulence simulations with an improved IPDG method and SST k-ω model have been conducted in the present study. In detail, the improved IPDG method was first introduced to eliminate the fourth-order homogeneity tensor connecting the viscous terms and variable gradients. The numerical accuracy has been systematically validated with several canonical cases. To overcome the numerical instability problem coming along with turbulence model simulation, a hybrid approach was implemented. Actually, finite volume method was utilized here for SST k-ω model. Then, the accuracy and reliability of the improved IPDG method have been analyzed in accompany with SST k-ω model and its compressibility correction. Actually, simulations of subsonic, transonic and supersonic cases have been conducted, including plate/airfoil boundary layers and shear layers. The accuracy has been comprehensively validated through quantitative comparison with experimental data, while the superiority of the compressibility correction could be observed at high Mach number.

$$hp$$-Optimal Interior Penalty Discontinuous Galerkin Methods for the Biharmonic Problem

We prove $$hp$$ -optimal error estimates for interior penalty discontinuous Galerkin methods (IPDG) for the biharmonic problem with homogeneous essential boundary conditions. We consider tensor product-type meshes in two and three dimensions, and triangular meshes in two dimensions. An essential ingredient in the analysis is the construction of a global $$H^2$$ piecewise polynomial approximants with $$hp$$ -optimal approximation properties over the given meshes. The $$hp$$ -optimality is also discussed for $$\mathcal C^0$$ -IPDG in two and three dimensions, and the stream formulation of the Stokes problem in two dimensions. Numerical experiments validate the theoretical predictions and reveal that $$p$$ -suboptimality occurs in presence of singular essential boundary conditions.