Discontinuous Finite Element Solution Research Articles

Error Approximation of the Second Order Hyperbolic Differential Equationby Using DG Finite Element Method

This article presents a simple efficient and asynchronously correcting a posteriori error approximation for discontinuous finite element solutions of the second-order hyperbolic partial differential problems on triangular meshes. This study considersthe basis functions for error spaces corresponding to some finite element spaces. The discretization error of each triangle is estimated by solving the local error problem. It also shows global super convergence for discontinuous solution on triangular lattice. In this article, the triangular elements are classify into three types: (i) elements with one inflow and two outflow edges are of type I, (ii) elements with two inflows and one outflow edges are of type II and (iii) elements with one inflow edge, one outflow edge, and one edge parallel to the characteristics are of type III. The article investigated higher-dimension discontinuous Galerkin methods for hyperbolic problems on triangular meshes and also studied the effect of finite element spaces on the superconvergence properties of DG solutions on three types of triangular elements and it showed that the DG solution is <i>O(h<sup>p+2</sup>)</i> superconvergent at Legendre points on the outflow edge on triangles having one outflow edge using three polynomial spaces. A posteriori error estimates are tested on a number of linear and nonlinear problems to show their efficiency and accuracy under lattice refinement for smooth and discontinuous solutions.

Using Deep Neural Networks for Detecting Spurious Oscillations in Discontinuous Galerkin Solutions of Convection-Dominated Convection–Diffusion Equations

Standard discontinuous Galerkin finite element solutions to convection-dominated convection–diffusion equations usually possess sharp layers but also exhibit large spurious oscillations. Slope limiters are known as a post-processing technique to reduce these unphysical values. This paper studies the application of deep neural networks for detecting mesh cells on which slope limiters should be applied. The networks are trained with data obtained from simulations of a standard benchmark problem with linear finite elements. It is investigated how they perform when applied to discrete solutions obtained with higher order finite elements and to solutions for a different benchmark problem.

Discontinuous finite element solutions for coupled radiation-conduction heat transfer in irregular media

The discontinuous finite element method (DFEM) is used to investigate the coupled radiation-conduction heat transfer in an irregular medium, and the highly accurate solutions for several typical media are numerically obtained. Comparing with the traditional continuous finite element method, the computational domain in the DFEM application is discretized into unstructured meshes that are assumed to be separated from each other. The shape function construction, field variable approximation, and numerical solutions are obtained for every single element. The continuity of the computational domain is maintained by modeling a numerical flux with the up-winding scheme. Thus the DFEM has the salient feature of geometry flexibility and simultaneously supports locally conservative solutions. The DFEM discretization for the radiative transfer equation and the energy diffusion equation are first presented, and the accuracies of the DFEM for coupled radiation-conduction heat transfer problems are verified. Combined radiation-conduction heat transfer problems in several irregular media are afterward solved, and the highly accurate DFEM solutions are presented.

Reduced order optimal control of the convective FitzHugh–Nagumo equations

In this paper, we compare three model order reduction methods: the proper orthogonal decomposition (POD), discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD) for the optimal control of the convective FitzHugh–Nagumo (FHN) equations. The convective FHN equations consist of the semi-linear activator and the linear inhibitor equations, modeling blood coagulation in moving excitable media. The semilinear activator equation leads to a non-convex optimal control problem (OCP). The most commonly used method in reduced optimal control is POD. We use DEIM and DMD to approximate efficiently the nonlinear terms in reduced order models. We compare the accuracy and computational times of three reduced-order optimal control solutions with the full order discontinuous Galerkin finite element solution of the convection dominated FHN equations with terminal controls. Numerical results show that POD is the most accurate whereas POD–DMD is the fastest.

High-order discontinuous Galerkin methods for coastal hydrodynamics applications

This paper applies high-order discontinuous Galerkin finite element solutions of the shallow water equations to realistic coastal hydrodynamics problems. The goal is to obtain results on par with the accuracy of current low-order models, while significantly surpassing their performance. Accomplishing this requires developing mesh discretizations with boundary and parameter field representations that are consistent with the high-order accuracy of the numerical methods. These developments allow coarse-mesh, high-order, solutions to provide comparable accuracy to today’s state-of-the-art high-resolution, low-order coastal models at reduced computational expense.

A reliable, efficient and localized error estimator for a discontinuous Galerkin method for the Signorini problem

We present a new residual-type a posteriori error estimator for the discontinuous finite element solution of contact problems. The theoretical results are derived for two and three-dimensional domains and arbitrary gap functions. The estimator yields upper and lower bounds to a suitable error notion which measures the error in the displacements and in a quantity related to the contact stresses and the actual contact zone. In the derivation of the error estimator the local properties of the discontinuous solution are exploited appropriately so that, on the one hand, the error estimator has no contributions related to the non-linearity in the interior of the actual contact zone and, on the other hand, the critical region between the actual and non-actual contact zone can be well refined.

Nonnegative Methods for Bilinear Discontinuous Differencing of the SN Equations on Quadrilaterals

Historically, matrix lumping and ad hoc flux fixups have been the only methods used to eliminate or suppress negative angular flux solutions associated with the unlumped bilinear discontinuous (UBLD) finite element spatial discretization of the two-dimensional SN equations. Though matrix lumping inhibits negative angular flux solutions of the SN equations, it does not guarantee strictly positive solutions. In this paper, we develop and define a strictly nonnegative, nonlinear, Petrov-Galerkin finite element method that fully preserves the bilinear discontinuous spatial moments of the transport equation. Additionally, we define two ad hoc fixups that maintain particle balance and explicitly set negative nodes of the UBLD finite element solution to zero but use different auxiliary equations to fully define their respective solutions.We assess the ability to inhibit negative angular flux solutions and the accuracy of every spatial discretization that we consider using a glancing void test problem with a discontinuous solution known to stress numerical methods. Though significantly more computationally intense, the nonlinear Petrov-Galerkin scheme results in a strictly nonnegative solution and is a more accurate solution than all the other methods considered. One fixup, based on shape preserving, results in a strictly nonnegative final solution but has increased numerical diffusion relative to the Petrov-Galerkin scheme and is less accurate than the UBLD solution. The second fixup, which preserves as many spatial moments as possible while setting negative values of the unlumped solution to zero, is less accurate than the Petrov-Galerkin scheme but is more accurate than the other fixup. However, it fails to guarantee a strictly nonnegative final solution. The fully lumped bilinear discontinuous finite element solution is the least accurate method, with significantly more numerical diffusion than the Petrov-Galerkin scheme and both fixups.

Superconvergence of bi-k Degree Time-Space Fully Discontinuous Finite Element for First-Order Hyperbolic Equations

AbstractIn this paper, we present a superconvergence result for the bi-k degree time-space fully discontinuous finite element of first-order hyperbolic problems. Based on the element orthogonality analysis (EOA), we first obtain the optimal convergence order of discontinuous Galerkin finite element solution. Then we use orthogonality correction technique to prove a superconvergence result at right Radau points, which is higher one order than the optimal convergence rate. Finally, numerical results are presented to illustrate the theoretical analysis.

Discontinuous finite element solution of the radiation diffusion equation on arbitrary polygonal meshes and locally adapted quadrilateral grids

In this paper, we propose a piece-wise linear discontinuous (PWLD) finite element discretization of the diffusion equation for arbitrary polygonal meshes. It is based on the standard diffusion form and uses the symmetric interior penalty technique, which yields a symmetric positive definite linear system matrix. A preconditioned conjugate gradient algorithm is employed to solve the linear system. Piece-wise linear approximations also allow a straightforward implementation of local mesh adaptation by allowing unrefined cells to be interpreted as polygons with an increased number of vertices. Several test cases, taken from the literature on the discretization of the radiation diffusion equation, are presented: random, sinusoidal, Shestakov, and Z meshes are used. The last numerical example demonstrates the application of the PWLD discretization to adaptive mesh refinement.

ElVis: A System for the Accurate and Interactive Visualization of High-Order Finite Element Solutions.

This paper presents the Element Visualizer (ElVis), a new, open-source scientific visualization system for use with high-order finite element solutions to PDEs in three dimensions. This system is designed to minimize visualization errors of these types of fields by querying the underlying finite element basis functions (e.g., high-order polynomials) directly, leading to pixel-exact representations of solutions and geometry. The system interacts with simulation data through runtime plugins, which only require users to implement a handful of operations fundamental to finite element solvers. The data in turn can be visualized through the use of cut surfaces, contours, isosurfaces, and volume rendering. These visualization algorithms are implemented using NVIDIA's OptiX GPU-based ray-tracing engine, which provides accelerated ray traversal of the high-order geometry, and CUDA, which allows for effective parallel evaluation of the visualization algorithms. The direct interface between ElVis and the underlying data differentiates it from existing visualization tools. Current tools assume the underlying data is composed of linear primitives; high-order data must be interpolated with linear functions as a result. In this work, examples drawn from aerodynamic simulations-high-order discontinuous Galerkin finite element solutions of aerodynamic flows in particular-will demonstrate the superiority of ElVis' pixel-exact approach when compared with traditional linear-interpolation methods. Such methods can introduce a number of inaccuracies in the resulting visualization, making it unclear if visual artifacts are genuine to the solution data or if these artifacts are the result of interpolation errors. Linear methods additionally cannot properly visualize curved geometries (elements or boundaries) which can greatly inhibit developers' debugging efforts. As we will show, pixel-exact visualization exhibits none of these issues, removing the visualization scheme as a source of uncertainty for engineers using ElVis.

A Superconvergent Local Discontinuous Galerkin Method for Elliptic Problems

In this manuscript we investigate the convergence properties of a minimal dissipation local discontinuous Galerkin(md-LDG) method for two-dimensional diffusion problems on Cartesian meshes. Numerical computations show O(h p+1) $\mathcal{L}^{2}$ convergence rates for the solution and its gradient and O(h p+2) superconvergent solutions at Radau points on enriched p-degree polynomial spaces. More precisely, a local error analysis reveals that the leading term of the LDG error for a p-degree discontinuous finite element solution is spanned by two (p+1)-degree right Radau polynomials in the x and y directions. Thus, LDG solutions are superconvergent at right Radau points obtained as a tensor product of the shifted roots of the (p+1)-degree right Radau polynomial. For tensor product polynomial spaces, the first component of the solution's gradient is O(h p+2) superconvergent at tensor product of the roots of left Radau polynomial in x and right Radau polynomial in y while the second component is O(h p+2) superconvergent at the tensor product of the roots of the right Radau polynomial in x and left Radau polynomial in y. Several numerical simulations are performed to validate the theory.

Ultraconvergence for averaging discontinuous finite elements and its applications in Hamiltonian system

This paper discusses the k-degree averaging discontinuous finite element solution for the initial value problem of ordinary differential equations. When k is even, the averaging numerical flux (the average of left and right limits for the discontinuous finite element at nodes) has the optimal-order ultraconvergence 2k + 2. For nonlinear Hamiltonian systems (e.g., Schrodinger equation and Kepler system) with momentum conservation, the discontinuous finite element methods preserve momentum at nodes. These properties are confirmed by numerical experiments.

FEM‐BEM coupling for the exterior Stokes problem with non‐conforming finite elements and an application to small droplet deformation dynamics

AbstractA non‐conforming, discontinuous Galerkin finite element–boundary element coupling procedure is presented for the exterior planar Stokes problem. The novel coupled formulation is developed using that for the conforming case as a guide to the introduction of extra mortar variables used to couple a discontinuous interior finite element solution with a continuous exterior boundary element solution. Convergence results for the new scheme are presented, for a range of different interior penalties, on computational domains discretized with regular structured meshes.To illustrate an application, the excitations required to model two‐phase droplet deformations in an extensional flow, under simple surface tension, with the new scheme are also presented. For a selection of different drop viscocities and exterior flows, with and without a rotational component, the progression to a steady‐state deformation of initially undeformed circular drops is calculated and the results compared with those from both a conforming FEM‐BEM equivalent scheme and from a small perturbation analysis where available. Copyright © 2011 John Wiley & Sons, Ltd.

A discontinuous finite element solution of the Boltzmann kinetic equation in collisionless and BGK forms for macroscopic gas flows

In this paper, an alternative approach to the traditional continuum analysis of flow problems is presented. The traditional methods, that have been popular with the CFD community in recent times, include potential flow, Euler and Navier–Stokes solvers. The method presented here involves solving the governing equation of the molecular gas dynamics that underlies the macroscopic behaviour described by the macroscopic governing equations. The equation solved is the Boltzmann kinetic equation in its simplified collisionless and BGK forms. The algorithm used is a discontinuous Taylor–Galerkin type and it is applied to the 2D problems of a highly rarefied gas expanding into a vacuum, flow over a vertical plate, rarefied hypersonic flow over a double ellipse, and subsonic and transonic flow over an aerofoil. The benefit of this type of solver is that it is not restricted to continuum regime (low Knudsen number) problems. However, it is a computationally expensive technique.

Superconvergence of local discontinuous Galerkin methods for one-dimensional convection–diffusion equations

In this paper, we study the convergence behavior of the local discontinuous Galerkin (LDG) methods when applied to one-dimensional time dependent convection–diffusion equations. We show that the LDG solution will be superconvergent towards a particular projection of the exact solution, if this projection is carefully chosen based on the convection and diffusion fluxes. The order is observed to be at least k + 2 when piecewise P k polynomials are used. Moreover, the numerical traces for the solution are also superconvergent, sometimes, of higher-order. This is a continuation of our previous work [Cheng Y, Shu C-W. Superconvergence and time evolution of discontinuous Galerkin finite element solutions. J Comput Phys 2008;227:9612–27], in which superconvergence of DG schemes for convection equations is discussed.

Superconvergence and time evolution of discontinuous Galerkin finite element solutions

In this paper, we study the convergence and time evolution of the error between the discontinuous Galerkin (DG) finite element solution and the exact solution for conservation laws when upwind fluxes are used. We prove that if we apply piecewise linear polynomials to a linear scalar equation, the DG solution will be superconvergent towards a particular projection of the exact solution. Thus, the error of the DG scheme will not grow for fine grids over a long time period. We give numerical examples of P k polynomials, with 1 ⩽ k ⩽ 3, to demonstrate the superconvergence property, as well as the long time behavior of the error. Nonlinear equations, one-dimensional systems and two-dimensional equations are numerically investigated to demonstrate that the conclusions hold true for very general cases.

The Discontinuous Galerkin Method for Two-dimensional Hyperbolic Problems Part II: A Posteriori Error Estimation

In this manuscript we construct simple, efficient and asymptotically correct a posteriori error estimates for discontinuous finite element solutions of scalar first-order hyperbolic partial differential problems on triangular meshes. We explicitly write the basis functions for the error spaces corresponding to several finite element spaces. The leading term of the discretization error on each triangle is estimated by solving a local problem. We also show global superconvergence for discontinuous solutions on triangular meshes. The a posteriori error estimates are tested on several linear and nonlinear problems to show their efficiency and accuracy under mesh refinement for smooth and discontinuous solutions.

The Discontinuous Galerkin Method for Two-Dimensional Hyperbolic Problems. Part I: Superconvergence Error Analysis

In this paper we investigate the superconvergence properties of the discontinuous Galerkin method applied to scalar first-order hyperbolic partial differential equations on triangular meshes. We show that the discontinuous finite element solution is O(h p+2) superconvergent at the Legendre points on the outflow edge for triangles having one outflow edge. For triangles having two outflow edges the finite element error is O(h p+2) superconvergent at the end points of the inflow edge. Several numerical simulations are performed to validate the theory. In Part II of this work we explicitly write down a basis for the leading term of the error and construct asymptotically correct a posteriori error estimates by solving local hyperbolic problems with no boundary conditions on more general meshes.

Superconvergence of discontinuous Galerkin solutions for a nonlinear scalar hyperbolic problem

In this paper we study the superconvergence of the discontinuous Galerkin solutions for nonlinear hyperbolic partial differential equations. On the first inflow element we prove that the p-degree discontinuous finite element solution converges at Radau points with an O( h p+2 ) rate. We further show that the solution flux converges on average at O( h 2 p+2 ) on element outflow boundary when no reaction terms are present. For reaction–convection problems we establish an O( h min(2 p+2, p+4) ) superconvergence rate of the flux on element outflow boundary. Globally, we prove that the flux converges at O( h 2 p+1 ) on average at the outflow of smooth-solution regions for nonlinear conservation laws. Numerical computations indicate that our results extend to nonrectangular meshes and nonuniform polynomial degrees. We further include a numerical example which shows that discontinuous solutions are superconvergent to the unique entropy solution away from shock discontinuities.

Superconvergence of Discontinuous Finite Element Solutions for Transient Convection–diffusion Problems

We present a study of the local discontinuous Galerkin method for transient convection-diffusion problems in one dimension. We show that p-degree piece-wise polynomial discontinuous finite element solutions of convection-dominated problems are O(Δxp+2) superconvergent at Radau points. For diffusion-dominated problems, the solution's derivative is O(Δxp+2) superconvergent at the roots of the derivative of Radau polynomial of degree p+1. Using these results, we construct several asymptotically exact a posteriori finite element error estimates. Computational results reveal that the error estimates are asymptotically exact.