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PAVEL BOCHEV
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Compatible discretizations

Compatible 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.

Collaborators

Resources

Intrepid
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.

Intrepid team:

Links

  • MSTK, mesh framework for unstructured 3D arbitrary topology meshes
  • GMV
  • TetGen

Atomistic-to-Continuum coupling

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 Pavel Bochev. 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.

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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

Domain Bridging

Currently, 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

We 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

Stabilized methods


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 formulation and analysis of a new class of stabilized methods for incompressible flows based on polynomial presssure projections. We also extended this class of methods to the Darcy flow problem.

Least-squares methods

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 residuals.

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Pavel Bochev — Distinguished Member of the Technical Staff.


Contact
E-mail: Pavel Bochev
(505)844-1990 (Phone)
(505)845-7442 (Fax)


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Sandia National Laboratories
P.O. Box 5800, MS 1320
Albuquerque, NM 87185-1320

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Sandia National Laboratories
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