Interior Penalty Discontinuous Galerkin Method Research Articles

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.

Families of hybridized interior penalty discontinuous Galerkin methods for locally degenerate advection-diffusion-reaction problems

We analyze families of primal high-order hybridized discontinuous Galerkin (HDG) methods for solving degenerate second-order elliptic problems. One major problem regarding this class of PDEs concerns its mathematical nature, which may be nonuniform over the whole domain. Due to the local degeneracy of the diffusion term, it can be purely hyperbolic in a subregion and elliptical in the rest. This problem is thus quite delicate to solve since the exact solution can be discontinuous at interfaces separating the elliptic and hyperbolic parts. The proposed hybridized interior penalty DG (H-IP) method is developed in a unified and compact fashion. It can handle pure-diffusive or -advective regimes as well as intermediate regimes that combine these two mechanisms for a wide range of Péclet numbers, including the tricky case of local evanescent diffusivity. To this end, an adaptive stabilization strategy based on the addition of jump-penalty terms is considered. An upwind-based scheme using a Lax–Friedrichs correction is favored for the hyperbolic region, and a Scharfetter–Gummel-based technique is preferred for the elliptic one. The well-posedness of the H-IP method is also briefly discussed by analyzing the strong consistency and the discrete coercivity condition in a self-adaptive energy-norm that is regime dependent. Extensive numerical experiments are performed to verify the model’s robustness for all the abovementioned regimes.

Discontinuous Galerkin approximations to elliptic and parabolic problems with a Dirac line source

The analyses of interior penalty discontinuous Galerkin methods of any order k for solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence of the method by deriving a priori error estimates in the L2 norm and in weighted energy norms. In addition, we prove almost optimal local error estimates in the energy norm for any approximation order. Further, almost optimal local error estimates in the L2 norm are obtained for the case of piecewise linear approximations whereas suboptimal error bounds in the L2 norm are shown for any polynomial degree. For the time-dependent case, convergence of semi-discrete and of backward Euler fully discrete scheme is established by proving error estimates in L2 in time and in space. Numerical results for the elliptic problem are added to support the theoretical results.

C 0-hybrid high-order methods for biharmonic problems

Abstract We devise and analyze $C^0$-conforming hybrid high-order (HHO) methods to approximate biharmonic problems with either clamped or simply supported boundary conditions. $C^0$-conforming HHO methods hinge on cell unknowns that are $C^0$-conforming polynomials of order $(k+2)$ approximating the solution in the mesh cells and on face unknowns, which are polynomials of order $k\ge 0$ approximating the normal derivative of the solution on the mesh skeleton. Such methods deliver $O(h^{k+1})$$H^2$-error estimates for smooth solutions. An important novelty in the error analysis is to lower the minimal regularity requirement on the exact solution. The technique to achieve this has a broader applicability than just $C^0$-conforming HHO methods, and to illustrate this point, we outline the error analysis for the well-known $C^0$-conforming interior penalty discontinuous Galerkin methods as well. The present technique does not require a $C^1$-smoother to evaluate the right-hand side in case of rough loads; loads in $W^{-1,q}$, $q>\frac {2d}{d+2}$, are covered, but not in $H^{-2}$. Finally, numerical results including comparisons to various existing methods showcase the efficiency of the proposed $C^0$-conforming HHO methods.

Error estimates for a linear folding model

Abstract An interior penalty discontinuous Galerkin method is devised to approximate minimizers of a linear folding model by discontinuous isoparametric finite element functions that account for an approximation of a folding arc. The numerical analysis of the discrete model includes an a priori error estimate in case of an accurate representation of the folding curve by the isoparametric mesh. Additional estimates show that geometric consistency errors may be controlled separately if the folding arc is approximated by piecewise polynomial curves. Various numerical experiments are carried out to validate the a priori error estimate for the folding model.

Improved a priori error estimates for a discontinuous Galerkin pressure correction scheme for the Navier–Stokes equations

AbstractThe pressure correction scheme is combined with interior penalty discontinuous Galerkin method to solve the time‐dependent Navier–Stokes equations. Optimal error estimates are derived for the velocity in the L2 norm in time and in space. Error bounds for the discrete time derivative of the velocity and for the pressure are also established. The analysis is challenging and technical and is based on appropriate use of lift operators and duality arguments.

A two‐grid discontinuous Galerkin method for an incompressible miscible displacement problem

AbstractAn incompressible miscible displacement problem in porous media is studied. A two‐grid algorithm based on an interior penalty discontinuous Galerkin method is proposed. With this algorithm, solving a nonlinear system on the discontinuous finite element space is reduced to solving a nonlinear problem on a coarse grid and a linear problem on a fine grid. The error estimate for the concentration in H1‐norm and the error estimate for the velocity in L2‐norm are obtained. The numerical experiments are provided to confirm our theoretical analysis.