Graham F. Carey

libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations

In this paper we describe the libMesh (http://libmesh.sourceforge.net) framework for parallel adaptive finite element applications. libMesh is an open-source software library that has been developed to facilitate serial and parallel simulation of multiscale, multiphysics applications using adaptive mesh refinement and coarsening strategies. The main software development is being carried out in the CFDLab (http://cfdlab.ae.utexas.edu) at the University of Texas, but as with other open-source software projects; contributions are being made elsewhere in the US and abroad. The main goals of this article are: (1) to provide a basic reference source that describes libMesh and the underlying philosophy and software design approach; (2) to give sufficient detail and references on the adaptive mesh refinement and coarsening (AMR/C) scheme for applications analysts and developers; and (3) to describe the parallel implementation and data structures with supporting discussion of domain decomposition, message passing, and details related to dynamic repartitioning for parallel AMR/C. Other aspects related to C++ programming paradigms, reusability for diverse applications, adaptive modeling, physics-independent error indicators, and similar concepts are briefly discussed. Finally, results from some applications using the library are presented and areas of future research are discussed.

Least-squares mixed finite elements for second-order elliptic problems

A theoretical analysis of a least-squares mixed finite element method for second-order elliptic problems in two- and three-dimensional domains is presented. It is proved that the method is not subj...

HYPERBOLIC HEAT TRANSFER WITH REFLECTION

The non-Fourier model for heat transfer leads to a hyperbolic evolution problem describing the temperature solution. A one-dimensional case is considered for such a propagating heat wave reflected at a boundary. A primary goal is investigation of the effectiveness of numerical solution techniques for the case of a propagating heat front and the influence of different boundary conditions. Finite elements are employed in space and alternative time integration schemes are studied, including ordinary differential equation system integrators. The effect of solution “roughness” on the error and oscillations before and after reflection is examined and rates of convergence are numerically determined.

A high-order compact formulation for the 3D Poisson equation

In this work we construct an extension to a class of higher-order compact methods for the three-dimensional Poisson equation. A superconvergent nodal rate of O(h6) is predicted, or O(h4) if the forcing function derivatives are not known exactly. Numerical experiments are conducted to verify these theoretical rates.

STREAM FUNCTION-VORTICITY DRIVEN CAVITY SOLUTION USING p FINITE ELEMENTS

Calculations for the two-dimensional driven cavity incompressible flow problem are presented. A p-type finite element scheme for the fully coupled stream function-vorticity formulation of the Navier-Stokes equations is used. Graded meshes are used to resolve vortex flow features and minimize the impact of corner singularities. Incremental continuation in the Reynolds number allows solutions to be computed for Re = 12 500. A significant feature of the work is that new tertiary and quaternary corner vortex features are observed in the flow field. Comparisons are made with other solutions in the literature.

Derivative calculation from finite element solutions

Abstract A technique is considered whereby very accurate derivatives of a finite element solution can be calculated efficiently. It is demonstrated here that most of the necessary quantities for this subsidiary computation are available as computed by-products in the preceding finite element solution procedure. The calculation is shown in this note to be a particular form of a procedure for which superconvergent theoretical error estimates have been proven elsewhere. Numerical experiments confirm the superior accuracy in the computed derivative (stress or flux).

Local refinement of simplicial grids based on the skeleton

In this paper we present a novel approach to the development of a class of local simplicial refinement strategies. The algorithm in two dimensions first subdivides certain edges. Then each triangle, if refined, is subdivided in two, three or four subelements depending on the previous division of its edges. Similarly, in three dimensions the algorithm begins by subdividing the two-dimensional triangulation composed by the faces of the tetrahedra (the skeleton) and then subdividing each tetrahedron in a compatible manner with the division of the faces. The complexity of the algorithm is linear in the number of added nodes. The algorithm is fully automatic and has been implemented to achieve global as well as local refinements. The numerical results obtained appear to confirm that the measure of degeneracy of subtetrahedra is bounded, and converges asymptotically to a fixed value when the refinement proceeds.

Approximate boundary-flux calculations☆

Abstract A technique for determining derivatives (fluxes or stresses) from finite element solutions is developed. The approach is a generalization to higher dimensions of a procedure known to give highly accurate results in one dimension. Numerical experiments demonstrate that certain difficulties are associated with corners in the higher-dimensional extensions and two variants of the method are examined. We consider both triangular and quadrilateral elements and observe some interesting differences in the numerical rates of convergence. Finally, this post-processing scheme is tested for nonlinear problems.

Analysis of spurious oscillation modes for the shallow water and Navier-Stokes equations

Abstract The origin and nature of spurious oscillation modes that appear in mixed finite element methods are examined. In particular, the shallow water equations are considered and a modal analysis for the one-dimensional problem is developed. From the resulting dispersion relations we find that the spurious modes in elevation are associated with zero frequency and large wave number (wavelengths of the order of the nodal spacing) and consequently are zero-velocity modes. The spurious modal behavior is the result of the finite spatial discretization. By means of an artificial compressibility and limiting argument we are able to resolve the similar problem for the Navier-Stokes equations. The relationship of this simpler analysis to alternative consistency arguments is explained. This modal approach provides an explanation of the phenomenon in question and permits us to deduce the cause of the very complex behavior of spurious modes observed in numerical experiments with the shallow water equations and Navier-Stokes equations. Furthermore, this analysis is not limited to finite element formulations, but is also applicable to finite difference formulations.

Penalty finite element method for the Navier-Stokes equations

Abstract We present an analysis of a penalty formulation of the stationary Navier-Stokes equations for an incompressible fluid. Subject to restrictions on the viscosity and prescribed body force, it is shown that there exists a unique solution to this penalty problem. The solution to the penalty problem is shown to converge to the solution of the Navier-Stokes problem as O(e) where e → 0 is the penalty parameter. Existence, uniqueness and stability properties for the approximate problem are then developed and we derive estimates for finite element approximation of the penalized Navier-Stokes problem presented here. Numerical studies are conducted to examine rates of convergence and sample numerical results presented for test cases.