Discontinuous Finite Element Solution Research Articles

Discontinuous finite element solution of 2-D radiative transfer with and without axisymmetry

A discontinuous Galerkin finite element methodology is presented for the computation of two-dimensional (2-D) radiative transfer in participating media with and without axisymmetry. The central idea of the discontinuous formulation is that the variables are allowed to be discontinuous across the inter-element boundaries. Consequently, the formulation should be particularly useful for radiative transfer calculations where the radiation intensity experiences a jump across the boundary. Being able to compute an axisymmetric problem over a 2-D mesh rather than using one three-dimensional (3-D) scenario results in reduced computing time and lower memory storage requirements. Mathematical formulations and numerical implementations using the discontinuous finite element (DFE) method for radiative transfer are given. The procedures for incorporating the mapping of the discontinuous formulation of radiative transfer develop a unified approach to both 2-D and 2-D axisymmetric problems are discussed in detail. The computed results are given, and they compare well with the solutions reported in the literature that were obtained using other methods. Examples include both non-scattering and scattering cases. The effects of the solid-angular and spatial discretization on the accuracy of results are discussed. Numerical simulations show that an even discretization of solid angle is important to ensure an adequate numerical accuracy.

A posteriori discontinuous finite element error estimation for two-dimensional hyperbolic problems

We analyze the discontinuous finite element errors associated with p-degree solutions for two-dimensional first-order hyperbolic problems. We show that the error on each element can be split into a dominant and less dominant component and that the leading part is O( h p+1 ) and is spanned by two ( p+1)-degree Radau polynomials in the x and y directions, respectively. We show that the p-degree discontinuous finite element solution is superconvergent at Radau points obtained as a tensor product of the roots of ( p+1)-degree Radau polynomial. For a linear model problem, the p-degree discontinuous Galerkin solution flux exhibits a strong O( h 2 p+2 ) local superconvergence on average at the element outflow boundary. We further establish an O( h 2 p+1 ) global superconvergence for the solution flux at the outflow boundary of the domain. These results are used to construct simple, efficient and asymptotically correct a posteriori finite element error estimates for multi-dimensional first-order hyperbolic problems in regions where solutions are smooth.

High-Order Accurate Discontinuous Finite Element Solution of the 2D Euler Equations

This paper deals with a high-order accurate discontinuous finite element method for the numerical solution of the Euler equations. The method combines two key ideas which are at the basis of the finite volume and of the finite element method, the physics of wave propagation being accounted for by means of Riemann problems and accuracy being obtained by means of high-order polynomial approximations within elements. We focus our attention on two-dimensional steady-state problems and present higher order accurate (up to fourth-order) discontinuous finite element solutions on unstructured grids of triangles. In particular we show that, in the presence of curved boundaries, a meaningful high-order accurate solution can be obtained only if a corresponding high-order approximation of the geometry is employed. We present numerical solutions of classical test cases computed with linear, quadratic, and cubic elements which illustrate the versatility of the method and the importance of the boundary condition

Discontinuous variational solutions for the neutron diffusion equation

A maximum variational principle derived by a least squares method is used to obtain discontinuous finite element solution to the neutron diffusion equation. Here the principle is applied with locally constant and locally linear trial functions to determine a discontinuous finite element solution for a number of one group diffusion problems. The results are in a good agreement with the analytical and the conventional continuous finite element solutions.

96/00345 Discontinuous finite element solutions for neutron transport in X-Y geometry

96/00345 Discontinuous finite element solutions for neutron transport in X-Y geometry

CONVERGENCE OF THE DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR HYPERBOLIC CONSERVATION LAWS

We prove convergence of the discontinuous Galerkin finite element method with polynomials of arbitrary degree q≥0 on general unstructured meshes for scalar conservation laws in multidimensions. We also prove for systems of conservation laws that limits of discontinuous Galerkin finite element solutions satisfy the entropy inequalities of the system related to convex entropies.

Discontinuous finite element solutions for neutron transport in X-Y geometry

A finite element method of composite solutions is described for the even-parity transport equation which uses a high order spherical harmonic expansions for the directional component of transport where they are needed and not everywhere as in established finite element methods. Previously in the method of composite solution trial functions were needed for both even and odd parity angular fluxes. For some problems there is a small saving in the number of unknowns used to specify a variational solution but for others savings of the order of 70% can be achived. The method is illustrated by numerical solutions in two dimensions for a Fletcher test problem, a PWR cell problem of Natelson and a problem of a quarter of a reactor with three control rods.

Propagation of error into regions of smoothness for non-linear approximations to hyperbolic equations

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.