Nonsymmetric Interior Penalty Galerkin Research Articles

Supercloseness of the NIPG method on a Bakhvalov-type mesh for a singularly perturbed problem with two small parameters

In this paper, the nonsymmetric interior penalty Galerkin (NIPG) method on a Bakhvalov-type mesh is proposed for a singularly perturbed problem with two small parameters. In order to reflect the behavior of layers more accurately, a balanced norm, rather than the common energy norm, is introduced. By selecting special penalty parameters at different mesh points, we establish the supercloseness of k+12 order, and prove an optimal order of uniform convergence in a balanced norm. Numerical experiments are proposed to confirm our theoretical results.

Supercloseness of the NIPG method for a singularly perturbed convection diffusion problem on Shishkin mesh in 2D

As a popular stabilization technique, the nonsymmetric interior penalty Galerkin (NIPG) method has significant application value in computational fluid dynamics. In this paper, we study the NIPG method for a typical two-dimensional singularly perturbed convection diffusion problem on a Shishkin mesh. According to the characteristics of the solution, the mesh and numerical scheme, a new composite interpolant is introduced. In fact, this interpolant is composed of a vertices-edges-element interpolant within the layer and a local L2-projection outside the layer. On the basis of that, by selecting penalty parameters on different types of interelement edges, we further obtain supercloseness of almost k+12 order in an energy norm. Here k is the degree of piecewise polynomials. Numerical tests support our theoretical conclusion.

Nonsymmetric interior penalty Galerkin method for nonlinear time-fractional integro-partial differential equations

This article discusses an efficient numerical method for solving nonlinear time-fractional integro-partial differential initial–boundary-value problems. To tackle the non-linearity of the problem, we first use the Newton linearization process. Then we apply the non-symmetric interior penalty Galerkin method for the spatial variable, and obtain a semi-discrete problem in the time variable. Finally, to obtain the fully-discrete scheme, we use both L1-scheme and L2-scheme for time-fractional derivative, and trapezoidal rule for integral term in the semi-discrete problem. We derive L2-norm stability and error estimates. The validity of the error analysis is supported by numerical experiments.

Supercloseness analysis of the nonsymmetric interior penalty Galerkin method for a singularly perturbed problem on Bakhvalov-type mesh

For a singularly perturbed convection diffusion problem, we study the supercloseness of the nonsymmetric interior penalty Galerkin (NIPG) method on a Bakhvalov-type mesh. In this process, a new composite interpolation is designed, which consists of Gauß Radau projection outside the layer and Gauß Lobatto projection inside the layer. Then by choosing the penalty parameters at different mesh points, we derive the supercloseness of k+12th order (k≥1) in an energy norm.

NIPG Finite Element Method for Convection-Dominated Diffusion Problems with Discontinuous Data

This paper presents the nonsymmetric interior penalty Galerkin (NIPG) finite element method for a class of one-dimensional convection dominated diffusion problems with discontinuous coefficients. The solution of the considered class of problem exhibits boundary and interior layers. Piecewise uniform Shishkin-type meshes are used for the spatial discretization. The error estimates in the energy norm have been derived for the proposed schemes. Theoretical results are supported by conducting numerical experiments. It is established that the errors are uniform with respect to the perturbation parameter [Formula: see text]. The uniformness of the error estimates with the perturbation parameter [Formula: see text] has also been established numerically for [Formula: see text]- norm.

Superconvergence error analysis of discontinuous Galerkin method with interior penalties for 2D elliptic convection – diffusion – reaction problems

This article focuses on constructing and analyzing a non-symmetric interior penalty Galerkin method (NIPG) on Shishkin mesh for solving singularly perturbed 2D elliptic boundary-value problems (BVPs). We use piecewise Lagrange interpolation at Gaussian points to improve the order of convergence of the interpolation error. We then study the superconvergence properties of the NIPG method and prove order of convergence in the discrete energy–norm. Various numerical experiments are provided to validate the theoretical results.

Supercloseness in a balanced norm of the NIPG method on Shishkin mesh for a reaction diffusion problem

For the error analysis of singularly perturbed reaction-diffusion problems, the balanced norm, which is stronger than the usual energy norm, is introduced to correctly reflect the behavior of the layer. In this paper, we study the convergence in a balanced norm for nonsymmetric interior penalty Galerkin (NIPG) method for the first time. For this purpose, a new interpolation is designed, which consists of a Gauß Lobatto interpolation in the layer, and a locally weighted L2 projection outside the layer. On that basis, by properly defining the penalty parameters at different nodes on a Shishkin mesh, we obtain the supercloseness of almost k+12 order, and prove the convergence of optimal order in a balanced norm. Here k is the degree of polynomials. Numerical experiments verify the main conclusion.

Non-Symmetric Interior Penalty Galerkin Finite Element Method for a Class of Singularly Perturbed Reaction Diffusion Problems with Discontinuous Data

Non-Symmetric Interior Penalty Galerkin Finite Element Method for a Class of Singularly Perturbed Reaction Diffusion Problems with Discontinuous Data

Study of the NIPG method for two–parameter singular perturbation problems on several layer adapted grids

In this paper, we apply the non-symmetric interior penalty Galerkin (NIPG) method to obtain the numerical solution of two-parameter singularly perturbed convection-diffusion-reaction boundary-value problems. In order to discretize the domain, here, we use the layer-adapted piecewise-uniform Shishkin mesh, the Bakhvalov mesh and the exponentially-graded mesh. We establish a superconvergence result of the NIPG method, that is, the proposed method is parameter-uniformly convergent with the order almost $$(k+1)$$ on the Shishkin mesh and $$(k+1)$$ on the Bakhvalov mesh and on the exponentially graded mesh in the energy norm, where k is the order of the polynomials. Numerical results comparing the three different types of meshes are presented at the end of the article supporting the theoretical error estimates.

The CG1–DG2 method for convection–diffusion equations in 2D

In this paper, we present the CG1–DG2 method for convection–diffusion equations. The space of continuous piecewise-linear functions is enriched with discontinuous quadratics so that the resultant finite element approximation is continuous at the vertices of the mesh but may have jumps across the edges. Three different approaches to the discretization of the diffusive part are considered: the symmetric interior penalty Galerkin method, the non-symmetric interior penalty Galerkin method and the Baumann–Oden method. In the context of elliptic problems we summarize well-known a priori error estimates for the discontinuous Galerkin approximation which carry over to the CG1–DG2 approach. Both methods have the same convergence rate which is also confirmed by numerical studies for diffusion and convection–diffusion problems.

Fully Computable Error Bounds for Discontinuous Galerkin Finite Element Approximations on Meshes with an Arbitrary Number of Levels of Hanging Nodes

We obtain fully computable a posteriori error bounds on the broken energy seminorm and discontinuous Galerkin norm (DG-norm) of the error in first order symmetric interior penalty Galerkin (SIPG), nonsymmetric interior penalty Galerkin (NIPG), and incomplete interior penalty Galerkin (IIPG) finite element approximations of a linear second order elliptic problem on meshes containing an arbitrary number of levels of hanging nodes and comprised of triangular elements. The estimators are completely free of unknown constants and provide guaranteed numerical bounds on the broken energy seminorm and DG-norm of the error. These estimators are also shown to provide a lower bound for the broken energy seminorm and DG-norm of the error up to a constant and higher order data oscillation terms. We also obtain an explicit computable bound for the value of the interior penalty parameter needed to ensure the existence of the discontinuous Galerkin finite element approximation for all versions of the method.

Constant free error bounds for nonuniform order discontinuous Galerkin finite-element approximation on locally refined meshes with hanging nodes

We obtain fully computable constant free a posteriori error bounds on the broken energy seminorm and the discontinuous Galerkin (DG) norm of the error for nonuniform polynomial order symmetric interior penalty Galerkin, nonsymmetric interior penalty Galerkin and incomplete interior penalty Galerkin finite-element approximations of a linear second-order elliptic problem on meshes containing hanging nodes and comprised of triangular elements. The estimators are completely free of unknown constants and provide guaranteed numerical bounds on the broken energy seminorm and the DG norm of the error. These estimators are also shown to provide a lower bound for the broken energy seminorm and the DG norm of the error up to a constant and higher-order data oscillation terms.

Sub-optimal Convergence of Non-symmetric Discontinuous Galerkin Methods for Odd Polynomial Approximations

We numerically verify that the non-symmetric interior penalty Galerkin method and the Oden-Babus?ka-Baumann method have sub-optimal convergence properties when measured in the L 2-norm for odd polynomial approximations. We provide numerical examples that use piece-wise linear and cubic polynomials to approximate a second-order elliptic problem in one and two dimensions.

On [formula omitted]-error estimate for nonsymmetric interior penalty Galerkin approximation to linear elliptic problems with nonhomogeneous Dirichlet data

In this paper, we present improved a priori error estimates for a nonsymmetric interior penalty Galerkin method (NIPG) with super-penalty for the problem − Δ u = f in Ω and u = g on ∂ Ω . Using piecewise polynomials of degree less than or equal to r , our new L 2 -error estimate is of order ( h / r ) r + 1 / 2 when g ∈ H r + 1 / 2 ( ∂ Ω ) and is optimal, i . e ., of order ( h / r ) r + 1 when g ∈ H r + 1 ( ∂ Ω ) , where h denotes the mesh size. Numerical experiments are presented to illustrate the theoretical results.

Local problem-based a posteriori error estimators for discontinuous Galerkin approximations of reactive transport

We consider adaptive discontinuous Galerkin (DG) methods for solving reactive transport problems in porous media. To guide anisotropic and dynamic mesh adaptation, a posteriori error estimators based on solving local problems are established. These error estimators are efficient to compute and effective to capture local phenomena, and they apply to all the four primal DG schemes, namely, symmetric interior penalty Galerkin, nonsymmetric interior penalty Galerkin, incomplete interior penalty Galerkin, and the Oden–Babuska-Baumann version of DG. Numerical results are provided to illustrate the effectiveness of the proposed error estimators.

A DYNAMIC, ADAPTIVE, LOCALLY CONSERVATIVE, AND NONCONFORMING SOLUTION STRATEGY FOR TRANSPORT PHENOMENA IN CHEMICAL ENGINEERING

A family of discontinuous Galerkin (DG) methods are formulated and applied to chemical engineering problems. They are the four primal discontinuous Galerkin schemes for space discretization: symmetric interior penalty Galerkin, Oden-Babuška-Baumann DG formulation, nonsymmetric interior penalty Galerkin, and incomplete interior penalty Galerkin. Numerical examples of DG to solve typical chemical engineering problems, including a diffusion-convection-reaction system in a catalytic particle, a problem of heat transfer in a fixed bed, and flow and contaminant transport simulations in porous media, are presented. This article highlights the substantial advantages of DG on adaptive mesh modification over traditional methods. In particular, we propose and investigate the dynamic mesh modification strategy for DG guided by mathematically sound a posteriori error estimators.

L2(H1 Norm A PosterioriError Estimation for Discontinuous Galerkin Approximations of Reactive Transport Problems

Explicita posteriori residual type error estimators in L2(H1) norm are derived for discontinuous Galerkin (DG) methods applied to transport in porous media with general kinetic reactions. They are flexible and apply to all the four primal DG schemes, namely, Oden--Babu?ka--Baumann DG (OBB-DG), non-symmetric interior penalty Galerkin (NIPG), symmetric interior penalty Galerkin (SIPG) and incomplete interior penalty Galerkin (IIPG). The error estimators use directly all the available information from the numerical solution and can be computed efficiently. Numerical examples are presented to demonstrate the efficiency and the effectivity of these theoretical estimators.

Symmetric and Nonsymmetric Discontinuous Galerkin Methods for Reactive Transport in Porous Media

For solving reactive transport problems in porous media, we analyze three primal discontinuous Galerkin (DG) methods with penalty, namely, symmetric interior penalty Galerkin (SIPG), nonsymmetric interior penalty Galerkin (NIPG), and incomplete interior penalty Galerkin (IIPG). A cut-off operator is introduced in DG to treat general kinetic chemistry. Error estimates in L2(H1) are established, which are optimal in h and nearly optimal in p. We develop a parabolic lift technique for SIPG, which leads to h-optimal and nearly p-optimal error estimates in the L2(L2) and negative norms. Numerical results validate these estimates. We also discuss implementation issues including penalty parameters and the choice of physical versus reference polynomial spaces.

A Posteriori error estimates for a discontinuous galerkin method applied to elliptic problems. Log number: R74

A posteriori error estimates for locally mass conservative methods for subsurface flow are presented. These methods are based on discontinuous approximation spaces and referred to as discontinuous Galerkin methods. In the case where penalty terms are added to the bilinear form, one obtains the nonsymmetric interior penalty Galerkin methods. In a previous work, we proved optimal rates of convergence of the methods applied to elliptic problems. Here, h adaptivity is investigated for flow problems in 2D. We derive global explicit estimators of the error in the L 2 norm and we numerically investigate an implicit indicator of the error in the energy norm. Model problems with discontinuous coefficients are considered.

Discontinuous and coupled continuous/discontinuous Galerkin methods for the shallow water equations

We consider the approximation of a simplified model of the depth-averaged two-dimensional shallow water equations by two approaches. In both approaches, a discontinuous Galerkin (DG) method is used to approximate the continuity equation. In the first approach, a continuous Galerkin method is used for the momentum equations. In the second approach a particular DG method, the nonsymmetric interior penalty Galerkin method, is used to approximate momentum. A priori error estimates are derived and numerical results are presented for both approaches.