Abani Patra

Problem decomposition for adaptive hp finite element methods

Problem decomposition strategies for load balancing parallel computations on adaptive hp finite element discretizations are discussed in this work. The special difficulties that arise in partitioning these discretizations are highlighted. Three classes of algorithms—mesh traversal based on orderings, interface based decompositions and recursive bisection of orderings are discussed. A new ordering scheme for efficient recursive bisection of orderings is introduced. Details of the algorithms and examples along with discussions of their merits and demerits are presented. Recursive bisection on the new ordering introduced here outperforms several known algorithms on test cases.

A parallel hp -adaptive discontinuous Galerkin method for hyperbolic conservation laws

This paper describes a parallel adaptive strategy based on discontinuous hp-finite element approximations of linear, scalar, hyperbolic conservation laws. The paper focuses on the development of an effective parallel adaptive strategy for such problems. Numerical experiments suggest that these techniques are highly parallelizable and deliver super-linear rates of convergence, thereby yielding efficiency many times superior to conventional schemes for hyperbolic problems.

A parallel adaptive strategy for hp finite element computations

Abstract In this work, a new three-step adaptive strategy for hp finite element discretizations is presented. Techniques for parallelizing these calculations on distributed memory multiprocessor computers are also developed. Numerical examples are used to demonstrate convergence properties and parallel efficiencies. Orders of magnitude reduction in computational efficiency over conventional methods are observed.

Parallel Domain Decomposition Solver for Adaptive hp Finite Element Methods

In this paper, the development and implementation of highly parallelizable domain decomposition solvers for adaptive hp finite element methods is discussed. Two-level orthogonalization is used to obtain a reduced system which is preconditioned by a coarse grid operator. The condition number of the preconditioned system, for Poisson problems in two space dimensions, is proved to be bounded by C(1 + Hp/h)2(1 + log p)2 and Cp(1 + Hp/h)2(1 + log p)2 for different choices of coarse grid operators, where H is the subdomain size, p is the maximum spectral order, h is the size of the smallest element in the subdomain, and C is a constant independent of the mesh parameters. The work here extends the work of Bramble et al. [Math Comp., 47 (1986), pp. 103--134] on the h-version and Babuska et al. [SIAM J. Numer. Anal., 29 (1991), pp. 624--661] on the p-version of the finite element method. A preliminary version of this solver was first announced by Oden, Patra, and Feng in [Domain Decomposition Solver for Adaptive hp Finite Elements, VII Conference on Domain Decomposition, State College, PA, October 1993]. Numerical experiments show fast convergence of the solver and good control of the condition number on a variety of discretizations.

Advances in studies of dense volcanic granular flows

The collapse and decrepitation of a lava dome at the summit of a volcano generally results in the generation of dense granular flows, often referred to as block and ash flows. As the dome particles propagate from the source, they break apart by internal pressure as well as collision. The propagation of block and ash flows can be simulated to some accuracy with a depth averaged numerical model of the equations of continuity and momentum for a material with a frictional resistance. However, important features of such flows, such as the influence of remote stress through force chains, erosion of the volcano substrate, and shock formation and pressurization upon particle break up are poorly understood. In the near future, the influence of these factors will be incorporated into depth averaged models. Various numerical techniques based on particles will some day yield results that can be compared not only with bulk flow properties, but to the internal layering of block and ash flow deposits.

Parallel adaptive hp finite element approximations for stokesian flows: Adaptive strategies, load balancing and domain decomposition solvers

Publisher Summary This chapter summarizes the development of a new class of algorithms using Parallel Adaptive hpFinite Elements for the analysis of Stokesian Flows. With parallel computing, these methods reduce computational costs associated with realistic finite element approximations. The chapter introduces the Stokes problem, its finite element formulation, and the appropriate function spaces necessary for a description of the various algorithms used in the solution process. It describes a simple adaptive strategy for producing good hpmeshes. The adaptive strategy comprises of three steps: selecting an intermediate error level between the initial mesh error and the final target mesh, and estimating different parameters; keeping polynomial orders PK constant; and keeping grid size constant and changing the local polynomial orders. The chapter also discusses a construction of compatible approximation spaces for the velocity and pressure spaces. It reviews a recursive load based bisection type mesh partitioning strategy. It describes a domain decomposition type solver for such problems.