Publications

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Optimal design, parameter estimation, and inverse problems arising in the modeling of semiconductor devices lead to optimization problems constrained by systems of PDEs. We study the impact of different state equation discretizations on optimization problems whose objective functionals involve flux terms. Galerkin methods, in which the flux is a derived quantity, are compared with mixed Galerkin discretizations where the flux is approximated directly. Our results show that the latter approach leads to more robust and accurate solutions of the optimization problem, especially for highly heterogeneous materials with large jumps in material properties. © 2008 Springer.

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This report is a collection of documents written as part of the Laboratory Directed Research and Development (LDRD) project A Mathematical Framework for Multiscale Science and Engineering: The Variational Multiscale Method and Interscale Transfer Operators. We present developments in two categories of multiscale mathematics and analysis. The first, continuum-to-continuum (CtC) multiscale, includes problems that allow application of the same continuum model at all scales with the primary barrier to simulation being computing resources. The second, atomistic-to-continuum (AtC) multiscale, represents applications where detailed physics at the atomistic or molecular level must be simulated to resolve the small scales, but the effect on and coupling to the continuum level is frequently unclear.

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We present a new family of stabilized methods for the Stokes problem. The focus of the paper is on the lowest order velocity-pressure pairs. While not LBB compliant, their simplicity and attractive computational properties make these pairs a popular choice in engineering practice. Our stabilization approach is motivated by terms that characterize the LBB "deficiency" of the unstable spaces. The stabilized methods are defined by using these terms to modify the saddle-point Lagrangian associated with the Stokes equations. The new stabilized methods offer a number of attractive computational properties. In contrast to other stabilization procedures, they are parameter free, do not require calculation of higher order derivatives or edge-based data structures, and always lead to symmetric linear systems. Furthermore, the new methods are unconditionally stable, achieve optimal accuracy with respect to solution regularity, and have simple and straightforward implementations. We present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods. © 2006 Society for Industrial and Applied Mathematics.

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Proliferation of degrees-of-freedom has plagued discontinuous Galerkin methodology from its inception over 30 years ago. This paper develops a new computational formulation that combines the advantages of discontinuous Galerkin methods with the data structure of their continuous Galerkin counterparts. The new method uses local, element-wise problems to project a continuous finite element space into a given discontinuous space, and then applies a discontinuous Galerkin formulation. The projection leads to parameterization of the discontinuous degrees-of-freedom by their continuous counterparts and has a variational multiscale interpretation. This significantly reduces the computational burden and, at the same time, little or no degradation of the solution occurs. In fact, the new method produces improved solutions compared with the traditional discontinuous Galerkin method in some situations. © 2005 Elsevier B.V. All rights reserved.