Publications

Siefert, Christopher S.; Tuminaro, Raymond S.; Bochev, Pavel B.; Boman, Erik G.; Devine, Karen D.

Abstract not provided.

Siefert, Christopher S.; Bochev, Pavel B.; Hu, Jonathan J.; Tuminaro, Raymond S.

Abstract not provided.

Ridzal, Denis R.; Bochev, Pavel B.

Abstract not provided.

Siefert, Christopher S.; Bochev, Pavel B.; Tuminaro, Raymond S.; Hu, Jonathan J.

Abstract not provided.

Bochev, Pavel B.; Collis, Samuel S.; Jones, Reese E.; Lehoucq, Richard B.; Parks, Michael L.; Scovazzi, Guglielmo S.; Silling, Stewart A.; Templeton, Jeremy A.

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.

Lehoucq, Richard B.; Bochev, Pavel B.

Abstract not provided.

Bochev, Pavel B.; Parks, Michael L.

Abstract not provided.

Tuminaro, Raymond S.; Siefert, Christopher S.; Bochev, Pavel B.

Abstract not provided.

Multiscale Modeling and Simulation

Bochev, Pavel B.; Parks, Michael L.

Abstract not provided.

Bochev, Pavel B.

Abstract not provided.

Lehoucq, Richard B.; Bochev, Pavel B.

Abstract not provided.

Bochev, Pavel B.

Abstract not provided.

Parks, Michael L.; Lehoucq, Richard B.; Bochev, Pavel B.

Abstract not provided.

Proposed for publication in SIAM Journal on Scientific Computing.

Bochev, Pavel B.; Tuminaro, Raymond S.; Hu, Jonathan J.

Abstract not provided.

Ridzal, Denis R.; Bochev, Pavel B.

Abstract not provided.

Siefert, Christopher S.; Bochev, Pavel B.; Hu, Jonathan J.; Tuminaro, Raymond S.

Abstract not provided.

SIAM Journal on Numerical Analysis

Bochev, Pavel B.; Dohrmann, Clark R.; Gunzburger, Max D.

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.

Tuminaro, Raymond S.; Bochev, Pavel B.; Hu, Jonathan J.

Abstract not provided.

Parks, Michael L.; Bochev, Pavel B.; Lehoucq, Richard B.

Abstract not provided.

Computer Methods in Applied Mechanics and Engineering

Hughes, Thomas J.R.; Scovazzi, Guglielmo S.; Bochev, Pavel B.; Buffa, Annalisa

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.

Scovazzi, Guglielmo S.; Bochev, Pavel B.; Collis, Samuel S.

Abstract not provided.

Day, David M.; Bochev, Pavel B.; Weiss, Chester J.; Robinson, Allen C.

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.

Proposed for publication in Methods and Applications of Analysis.

Bochev, Pavel B.

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.

Scovazzi, Guglielmo S.; Bochev, Pavel B.

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.

Bochev, Pavel B.

Abstract not provided.