me if you have questions about any of the research topics.
discretizations transform partial differential equations discrete algebraic problems that mimic fundamental properties of the continuum
equations. We provide a common framework for mimetic discretizations
using algebraic topology to guide our analysis. The framework and all attendant
discrete structures are put together by using two basic mappings between differential forms and cochains. The
key concept of the framework is a natural inner product on cochains which
induces a combinatorial Hodge theory on the cochain complex. The framework supports mutually
consistent operations of differentiation and integration, has a combinatorial Stokes theorem, and
preserves the invariants of the De Rham cohomology groups. This allows, among other things, 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. These methods result from a choice of the reconstruction
operator and are equivalent for some classes of reconstruction operators.
Interoperable Tools for Rapid Development of Compatible Discretizations
Multi-physics models with multiple temporal and spatial scales are critical for predictive simulations in key Sandia applications. Solution of these models poses insurmountable challenges to existing computational paradigms that are, for the most part, designed to deal with single-physics models; in the future, one cannot expect them to lead to radically improved simulations of these complex problems. This project aims to close this functionality and capability gap by developing a new generation of computational infrastructure consisting of interoperable and extendable software libraries, based on high-order accurate, compatible, and adaptive discretization methods.
The main goal of the Intrepid project is to develop a hierarchy of interoperable software tools that provide a basis for extendable software libraries for scientific computing. Intrepid consists of the following components:
- core toolset: a software realization of a discrete differential complex. Such a complex enables stable, accurate, efficient, and physically consistent numerical solution of the fundamental equations of mathematical physics by Galerkin, mixed Galerkin, mimetic and finite volume methods on arbitrary unstructured grids. This hitherto unavailable level of discretization support and functionality is achieved by providing interoperable implementations of the two key realizations of the De Rham complex: by a polynomial differential complex and by a mimetic (cochain) complex; see Principles of compatible discretizations.
- application support algorithms: extend the core toolset by providing enabling tools for application methods such as arbitrary Lagrangian-Eulerian methods; hybridized discontinuous Galerkin methods; variational multiscale; and least-squares principles. They include algorithms for conservative remap, multimaterial interface reconstruction, domain decomposition, mesh tying, adaptive error control and reduced order modeling.
- application methods: Intrepid tools will be used to demonstrate implementation of numerical methods for PDEs arising in select applications of interest to Sandia
In the design of the core toolset and the application support algorithms Intrepid follows the established and highly successful "package" design philosophy of Trilinos. A package is an integral unit developed by a team of experts which exists under a top level that provides a common configuration, documentation, licensing and bug-tracking.
- MSTK, mesh framework for unstructured 3D arbitrary topology meshes
The research on A Mathematical Analysis of Atomistic-to-Continuum
(AtC) Coupling Methods is funded by a three year DOE award,
starting fiscal year 2006. The project is a collaborative, multi-institutional
effort led by the Sandians
Rich Lehoucq and
Other participating institutions and senior personnel are:
The research is funded under the Office of Science's Multiscale
Mathematics program. The program addresses those science
problems that span many time scales--from femtoseconds to years--and
many length scales--from the atomic level to the macroscopic.
Atomistic-to-Continuum (AtC) coupling enables a continuum calculation to
be performed over the majority of a domain of interest while limiting
the more expensive atomistic simulation over a subset of the domain.
Unfortunately, resolving an atomistic and continuum view is difficult
because the former is based on non-local force interactions that occur
over a finite distance, in constrast to the former continuum view.
Carbon nanotubes —
a prototypical AtC application.
Stabilized, multiscale and DG methods
My research in multiscale methods includes domain bridging and application of
variational multiscale analysis to Discontinuous Galerkin methods
our main research focus is on mesh tying for domains with non-coincident
interfaces. This situation may arise when two domains sharing a common curved
interface are meshed independently. Mortar methods for mesh tying define
Lagrange multipliers on one of the two discrete interfaces, usually called
slave interface. Projection operators are used to transfer data from the other
interface and to set up the constraint. In this research we consider approaches
based on extension of least-squares methods for transmission problems and
novel, discrete Lagrange multiplier approaches that use interface perturbations
to pass a linear patch test. Our research group includes
Multiscale DG Methods
developed a new class of Discontinuous Galerkin (DG) methods based on variational multiscale
ideas. Our approach begins with an
additive decomposition of the discontinuous finite element space into
continuous (coarse) and discontinuous (fine) components. Variational multiscale
analysis is used to define an interscale transfer operator that associates
coarse and fine scale functions. Composition of this operator with a donor DG
method yields a new formulation that combines the advantages of DG methods with
the attractive and more efficient computational structure of a continuous
Galerkin method. The key to the success of the new approach is efficient
computation of the interscale operator. Variational Multiscale Analysis leads
to a natural definition of local, elementwise problems that mimic the structure
of the donor DG formulation. My collaborators are
Stable and accurate finite element solution of mixed variational equations requires
finite element spaces that verify the inf-sup, or LBB, stability condition. Stabilized methods
are a class of methods for mixed problems that enable the use of non LBB compliant spaces
by modification of the mixed variational form. Togehter with Clark Dohrmann
and Max Gunzburger I worked on
of a new class of stabilized methods for incompressible
flows based on polynomial presssure projections. We also
extended this class of methods to the
Least-squares finite element methods are an attractive class of methods
for the numerical solution of partial differential equations. They are
motivated by the desire to recover, in general settings, the
advantageous features of Rayleigh-Ritz methods such as the avoidance of
discrete compatibility conditions and the production of symmetric and
positive definite discrete systems. The methods are based on the
minimization of convex functionals that are constructed from equation
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Pavel Bochev — Distinguished Member of the Technical Staff.
E-mail: Pavel Bochev
Sandia National Laboratories
P.O. Box 5800, MS 1320
Albuquerque, NM 87185-1320
Sandia National Laboratories
1515 Eubank SE,
Albuquerque, NM 87123-1320