Victor M. Calo

A class of discontinuous Petrov-Galerkin methods. Part IV: The optimal test norm and time-harmonic wave propagation in 1D

The phase error, or the pollution effect in the finite element solution of wave propagation problems, is a well known phenomenon that must be confronted when solving problems in the high-frequency range. This paper presents a new method with no phase errors for one-dimensional (1D) time-harmonic wave propagation problems using new ideas that hold promise for the multidimensional case. The method is constructed within the framework of the discontinuous Petrov-Galerkin (DPG) method with optimal test functions. We have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm. In this paper, we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm. This is answered in the specific case of the Helmholtz equation with L^2-norm as the energy norm. We obtain uniform stability with respect to the wave number. We illustrate the method with a number of 1D numerical experiments, using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally. A 1D theoretical stability analysis is also developed.

The Cost of Continuity: Performance of Iterative Solvers on Isogeometric Finite Elements

In this paper we study how the use of a more continuous set of basis functions affects the cost of solving systems of linear equations resulting from a discretized Galerkin weak form. Specifically, we compare performance of linear solvers when discretizing using

A parallel direct solver for the self-adaptive hp Finite Element Method

C^0

Self-Assembled Asymmetric Block Copolymer Membranes: Bridging the Gap from Ultra- to Nanofiltration

B-splines, which span traditional finite element spaces, and

Mode decomposition methods for flows in high-contrast porous media. Global-local approach

C^{p-1}

Variational and Multiscale Methods in Turbulence

B-splines, which represent maximum continuity. We provide theoretical estimates for the increase in cost of the matrix-vector product as well as for the construction and application of black-box preconditioners. We accompany these estimates with numerical results and study their sensitivity to various grid parameters such as element size

Fast Multiscale Reservoir Simulations using POD-DEIM Model Reduction

h

An energy-stable convex splitting for the phase-field crystal equation

and polynomial order of approximation

Gaussian quadrature for splines via homotopy continuation

p

PetIGA-MF: A multi-field high-performance toolbox for structure-preserving B-splines spaces

. Finally, we present timing results for a range of preconditioning options for the Laplace problem. We conclude that the matrix-vector product operation is at most