Joseph E. Flaherty

Parallel, adaptive finite element methods for conservation laws

We construct parallel finite element methods for the solution of hyperbolic conservation laws in one and two dimensions. Spatial discretization is performed by a discontinuous Galerkin finite element method using a basis of piecewise Legendre polynomials. Temporal discretization utilizes a Runge-Kutta method. Dissipative fluxes and projection limiting prevent oscillations near solution discontinuities. A posteriori estimates of spatial errors are obtained by a p-refinement technique using superconvergence at Radau points. The resulting method is of high order and may be parallelized efficiently on MIMD computers. We compare results using different limiting schemes and demonstrate parallel efficiency through computations on an NCUBE/2 hypercube. We also present results using adaptive h- and p-refinement to reduce the computational cost of the method.

Adaptive Local Refinement with Octree Load Balancing for the Parallel Solution of Three-Dimensional Conservation Laws

Conservation laws are solved by a local Galerkin finite element procedure with adaptive space-time mesh refinement and explicit time integration. The Courant stability condition is used to select smaller time steps on smaller elements of the mesh, thereby greatly increasing efficiency relative to methods having a single global time step. Processor load imbalances, introduced at adaptive enrichment steps, are corrected by using traversals of an octree representing a spatial decomposition of the domain. To accommodate the variable time steps, octree partitioning is extended to use weights derived from element size. Partition boundary smoothing reduces the communications volume of partitioning procedures for a modest cost. Computational results comparing parallel octree and inertial partitioning procedures are presented for the three-dimensional Euler equations of compressible flow solved on an IBM SP2 computer.

A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems

Abstract We analyze the spatial discretization errors associated with solutions of one-dimensional hyperbolic conservation laws by discontinuous Galerkin methods (DGMs) in space. We show that the leading term of the spatial discretization error with piecewise polynomial approximations of degree p is proportional to a Radau polynomial of degree p +1 on each element. We also prove that the local and global discretization errors are O( Δx 2( p +1) ) and O( Δx 2 p +1 ) at the downwind point of each element. This strong superconvergence enables us to show that local and global discretization errors converge as O( Δx p +2 ) at the remaining roots of Radau polynomial of degree p +1 on each element. Convergence of local and global discretization errors to the Radau polynomial of degree p +1 also holds for smooth solutions as p →∞. These results are used to construct asymptotically correct a posteriori estimates of spatial discretization errors that are effective for linear and nonlinear conservation laws in regions where solutions are smooth.

An Adaptive Discontinuous Galerkin Technique with an Orthogonal Basis Applied to Compressible Flow Problems

We present a high-order formulation for solving hyperbolic conservation laws using the discontinuous Galerkin method (DGM). We introduce an orthogonal basis for the spatial discretization and use explicit Runge--Kutta time discretization. Some results of higher order adaptive refinement calculations are presented for inviscid Rayleigh--Taylor flow instability and shock reflection problems. The adaptive procedure uses an error indicator that concentrates the computational effort near discontinuities.

A moving finite element method with error estimation and refinement for one-dimensional time dependent partial differential equations

We discuss a moving finite element method for solving vector systems of time dependent partial differential equations in one space dimension. The mesh is moved so as to equidistribute the spatial component of the discretization error in

Parallel structures and dynamic load balancing for adaptive finite element computation

H^1

Geometry representation issues associated with p-version finite element computations

. We present a method of estimating this error by using p-hierarchic finite elements. The error estimate is also used in an adaptive mesh refinement procedure to give an algorithm that combines mesh movement and refinement.We discretize the partial differential equations in space using a Galerkin procedure with piecewise linear elements to approximate the solution and quadratic elements to estimate the error. A system of ordinary differential equations for mesh velocities are used to control element motions. We use existing software for stiff ordinary differential equations for the temporal integration of the solution, the error estimate, and the mesh motion. Computational results using a code based on our method are presented for several examples.

A moving-mesh finite element method with local refinement for parabolic partial differential equations

Abstract An adaptive technique for a partial differential system automatically adjusts a computational mesh or varies the order of a numerical procedure to obtain a solution satisfying prescribed accuracy criteria in an optimal fashion. We describe data structures for distributed storage of finite element mesh data as well as software for mesh adaptation, load balancing, and solving compressible flow problems. Processor load imbalances are introduced at adaptive enrichment steps during the course of a parallel computation. To correct this, we have developed three dynamic load balancing procedures based, respectively, on load imbalance trees, moment of inertia, and octree traversal. Computational results on an IBM SP2 computer are presented for steady and transient solutions of the three-dimensional Euler equations of compressible flow.

Load balancing for the parallel adaptive solution of partial differential equations

This paper addresses issues related to accurate geometry representation for p-version finite elements on curved three-dimensional domains. Specific options to account for domain geometry information during element-level computation are identified. Accuracy requirements on the geometry related approximations to preserve the optimal rate of finite element error convergence for second-order elliptic boundary value problems are given. An element geometric mapping scheme based on blending the exact shape of the domain boundary is described that can either be used directly during element integrations, or used to construct element-level geometric approximations of required accuracy. Smoothness issues of the rational blends on simplex topologies are discussed and a numerical example based on the solution of Poissons equation in three dimensions is presented to illustrate the impact of the rational blends on the optimal rate of finite element error convergence.

On the stability of mesh equidistribution strategies for time-dependent partial differential equations

Abstract We discuss a moving-mesh finite element method for solving initial boundary value problems for vector systems of partial differential equations in one space dimension and time. The system is discretized using piecewise linear finite element approximations in space and a backward difference code for stiff ordinary differential systems in time. A spatial-error estimation is calculated using piecewise quadratic approximations that use the superconvergence properties of parabolic systems to gain computational efficiency. The spatial-error estimate is used to move and locally refine the finite element mesh in order to equidistribute a measure of the total spatial error and to satisfy a prescribed error tolerance. Ordinary differential equations for the spatial-error estimate and the mesh motion are integrated in time using the same backward difference software that is used to determine the numerical solution of the partial differential system. We present several details of an algorithm that may be used to develop a general-purpose finite element code for one-dimensional parabolic partial differential systems. The algorithm combines mesh motion and local refinement in a relatively efficient manner and attempts to eliminate problemdependent numerical parameters. A variety of examples that motivate our mesh-moving strategy and illustrate the performance of our algorithm are presented.