Publications

Abstract not provided.

Electromagnetic induction is a classic geophysical exploration method designed for subsurface characterization--in particular, sensing the presence of geologic heterogeneities and fluids such as groundwater and hydrocarbons. Several approaches to the computational problems associated with predicting and interpreting electromagnetic phenomena in and around the earth are addressed herein. Publications resulting from the project include [31]. To obtain accurate and physically meaningful numerical simulations of natural phenomena, computational algorithms should operate in discrete settings that reflect the structure of governing mathematical models. In section 2, the extension of algebraic multigrid methods for the time domain eddy current equations to the frequency domain problem is discussed. Software was developed and is available in Trilinos ML package. In section 3 we consider finite element approximations of De Rham's complex. We describe how to develop a family of finite element spaces that forms an exact sequence on hexahedral grids. The ensuing family of non-affine finite elements is called a van Welij complex, after the work [37] of van Welij who first proposed a general method for developing tangentially and normally continuous vector fields on hexahedral elements. The use of this complex is illustrated for the eddy current equations and a conservation law problem. Software was developed and is available in the Ptenos finite element package. The more popular methods of geophysical inversion seek solutions to an unconstrained optimization problem by imposing stabilizing constraints in the form of smoothing operators on some enormous set of model parameters (i.e. ''over-parametrize and regularize''). In contrast we investigate an alternative approach whereby sharp jumps in material properties are preserved in the solution by choosing as model parameters a modest set of variables which describe an interface between adjacent regions in physical space. While still over-parametrized, this choice of model space contains far fewer parameters than before, thus easing the computational burden, in some cases, of the optimization problem. And most importantly, the associated finite element discretization is aligned with the abrupt changes in material properties associated with lithologic boundaries as well as the interface between buried cultural artifacts and the surrounding Earth. In section 4, algorithms and tools are described that associate a smooth interface surface to a given triangulation. In particular, the tools support surface refinement and coarsening. Section 5 describes some preliminary results on the application of interface identification methods to some model problems in geophysical inversion. Due to time constraints, the results described here use the GNU Triangulated Surface Library for the manipulation of surface meshes and the TetGen software library for the generation of tetrahedral meshes.

The approximate solution of optimization and control problems for systems governed by linear, elliptic partial differential equations is considered. Such problems are most often solved using methods based on the application of the Lagrange multiplier rule followed by discretization through, e.g., a Galerkin finite element method. As an alternative, we show how least-squares finite element methods can be used for this purpose. Penalty-based formulations, another approach widely used in other settings, have not enjoyed the same level of popularity in the partial differential equation case perhaps because naively defined penalty-based methods can have practical deficiencies. We use methodologies associated with modern least-squares finite element methods to develop and analyze practical penalty methods for the approximate solution of optimization problems for systems governed by linear, elliptic partial differential equations. We develop an abstract theory for such problems; along the way, we introduce several methods based on least-squares notions, and compare and constrast their properties.

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.

Abstract not provided.

We provide a common framework for compatible discretizations using algebraic topology to guide our analysis. The main concept is the natural inner product on cochains, which induces a combinatorial Hodge theory. The framework comprises of mutually consistent operations of differentiation and integration, has a discrete Stokes theorem, and preserves the invariants of the DeRham cohomology groups. The latter allows for an elementary calculation of the kernel of the discrete Laplacian. Our framework provides an abstraction that includes examples of compatible finite element, finite volume and finite difference methods. We describe how these methods result from the choice of a reconstruction operator and when they are equivalent.

Abstract not provided.

Least-squares finite-element methods for Darcy flow offer several advantages relative to the mixed-Galerkin method: the avoidance of stability conditions between finite-element spaces, the efficiency of solving symmetric and positive definite systems, and the convenience of using standard, continuous nodal elements for all variables. However, conventional C{sup o} implementations conserve mass only approximately and for this reason they have found limited acceptance in applications where locally conservative velocity fields are of primary interest. In this paper, we show that a properly formulated compatible least-squares method offers the same level of local conservation as a mixed method. The price paid for gaining favourable conservation properties is that one has to give up what is arguably the least important advantage attributed to least-squares finite-element methods: one can no longer use continuous nodal elements for all variables. As an added benefit, compatible least-squares methods inherit the best computational properties of both Galerkin and mixed-Galerkin methods and, in some cases, yield identical results, while offering the advantages of not having to deal with stability conditions and yielding positive definite discrete problems. Numerical results that illustrate our findings are provided.

Abstract not provided.

Abstract not provided.

Existing approaches in multiscale science and engineering have evolved from a range of ideas and solutions that are reflective of their original problem domains. As a result, research in multiscale science has followed widely diverse and disjoint paths, which presents a barrier to cross pollination of ideas and application of methods outside their application domains. The status of the research environment calls for an abstract mathematical framework that can provide a common language to formulate and analyze multiscale problems across a range of scientific and engineering disciplines. In such a framework, critical common issues arising in multiscale problems can be identified, explored and characterized in an abstract setting. This type of overarching approach would allow categorization and clarification of existing models and approximations in a landscape of seemingly disjoint, mutually exclusive and ad hoc methods. More importantly, such an approach can provide context for both the development of new techniques and their critical examination. As with any new mathematical framework, it is necessary to demonstrate its viability on problems of practical importance. At Sandia, lab-centric, prototype application problems in fluid mechanics, reacting flows, magnetohydrodynamics (MHD), shock hydrodynamics and materials science span an important subset of DOE Office of Science applications and form an ideal proving ground for new approaches in multiscale science.

Implicit time integration coupled with SUPG discretization in space leads to additional terms that provide consistency and improve the phase accuracy for convection dominated flows. Recently, it has been suggested that for small Courant numbers these terms may dominate the streamline diffusion term, ostensibly causing destabilization of the SUPG method. While consistent with a straightforward finite element stability analysis, this contention is not supported by computational experiments and contradicts earlier Von-Neumann stability analyses of the semidiscrete SUPG equations. This prompts us to re-examine finite element stability of the fully discrete SUPG equations. A careful analysis of the additional terms reveals that, regardless of the time step size, they are always dominated by the consistent mass matrix. Consequently, SUPG cannot be destabilized for small Courant numbers. Numerical results that illustrate our conclusions are reported.

Abstract not provided.

Abstract not provided.

Abstract not provided.

Abstract not provided.

Abstract not provided.