arXiv subjects

Michael Holst

Publications and source records attributed to Michael Holst.

48 records · Page 3Linked to original sources

arXiv2010

Applications of Domain Decomposition and Partition of Unity Methods in Physics and Geometry

We consider a class of adaptive multilevel domain decomposition-like algorithms, built from a combination of adaptive multilevel finite element, domain decomposition, and partition of unity methods. These algorithms have several interesting features such as very low communication requirements, and they inherit a simple and elegant approximation theory framework from partition of unity methods. They are also very easy to use with highly complex sequential adaptive finite element packages, requiring little or no modification of the underlying sequential finite element software. The parallel algorithm can be implemented as a simple loop which starts off a sequential local adaptive solve on a collection of processors simultaneously. We first review the Partition of Unity Method (PUM) of Babuvska and Melenk, and outline the PUM approximation theory framework. We then describe a variant we refer to here as the Parallel Partition of Unity Method (PPUM), which is a combination of the Partition of Unity Method with the parallel adaptive algorithm of Bank and Holst. We then derive two global error estimates for PPUM, by exploiting the PUM analysis framework it inherits, and by employing some recent local estimates of Xu and Zhou. We then discuss a duality-based variant of PPUM which is more appropriate for certain applications, and we derive a suitable variant of the PPUM approximation theory framework. Our implementation of PPUM-type algorithms using the FETK and MC software packages is described. We then present a short numerical example involving the Einstein constraints arising in gravitational wave models.

math.NA↗

arXiv2010

Adaptive Numerical Treatment of Elliptic Systems on Manifolds

Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element methods for approximating solutions to this class of problems are considered in some detail. Two a posteriori error indicators are derived, based on local residuals and on global linearized adjoint or dual problems. The design of Manifold Code (MC) is then discussed; MC is an adaptive multilevel finite element software package for 2- and 3-manifolds. It employs a posteriori error estimation, adaptive simplex subdivision, unstructured algebraic multilevel methods, global inexact Newton methods, and numerical continuation methods for the numerical solution of nonlinear covariant elliptic systems on 2- and 3-manifolds. Some of the more interesting features of MC are described in detail, including some new ideas for topology and geometry representation in simplex meshes, and an unusual partition of unity-based method for exploiting parallel computers. A short example is then given which involves the Hamiltonian and momentum constraints in the Einstein equations, a representative nonlinear 4-component covariant elliptic system on a Riemannian 3-manifold which arises in general relativity. A number of operator properties and solvability results recently established are first summarized, making possible two quasi-optimal a priori error estimates for Galerkin approximations which are then derived. These two results complete the theoretical framework for effective use of adaptive multilevel finite element methods. A sample calculation using the MC software is then presented.

math.NA↗

arXiv2010

Optimality of multilevel preconditioners for local mesh refinement in three dimensions

In this article, we establish optimality of the Bramble-Pasciak-Xu (BPX) norm equivalence and optimality of the wavelet modified (or stabilized) hierarchical basis (WHB) preconditioner in the setting of local 3D mesh refinement. In the analysis of WHB methods, a critical first step is to establish the optimality of BPX norm equivalence for the refinement procedures under consideration. While the available optimality results for the BPX norm have been constructed primarily in the setting of uniformly refined meshes, a notable exception is the local 2D red-green result due to Dahmen and Kunoth. The purpose of this article is to extend this original 2D optimality result to the local 3D red-green refinement procedure introduced by Bornemann-Erdmann-Kornhuber (BEK), and then to use this result to extend the WHB optimality results from the quasiuniform setting to local 2D and 3D red-green refinement scenarios. The BPX extension is reduced to establishing that locally enriched finite element subspaces allow for the construction of a scaled basis which is formally Riesz stable. It is possible to show that the number of degrees of freedom used for smoothing is bounded by a constant times the number of degrees of freedom introduced at that level of refinement, indicating that a practical implementable version of the resulting BPX preconditioner for the BEK refinement setting has provably optimal (linear) computational complexity per iteration. An interesting implication of the optimality of the WHB preconditioner is the a priori H1-stability of the L2-projection. The theoretical framework employed supports arbitrary spatial dimension d >= 1 and requires no coefficient smoothness assumptions beyond those required for well-posedness in H1.

math.NA↗

arXiv2010

Local Refinement and Multilevel Preconditioning: Implementation and Numerical Experiments

In this paper, we examine a number of additive and multiplicative multilevel iterative methods and preconditioners in the setting of two-dimensional local mesh refinement. While standard multilevel methods are effective for uniform refinement-based discretizations of elliptic equations, they tend to be less effective for algebraic systems which arise from discretizations on locally refined meshes, losing their optimal behavior in both storage and computational complexity. Our primary focus here is on BPX-style additive and multiplicative multilevel preconditioners, and on various stabilizations of the additive and multiplicative hierarchical basis method (HB), and their use in the local mesh refinement setting. In this article, we describe in detail the implementation of these types of algorithms, including detailed discussions of the datastructures and traversal algorithms we employ for obtaining optimal storage and computational complexity in our implementations. We show how each of the algorithms can be implemented using standard datatypes available in languages such as C and FORTRAN, so that the resulting algorithms have optimal (linear) storage requirements, and so that the resulting multilevel method or preconditioner can be applied with optimal (linear) computational costs. Our implementations are performed in both C and MATLAB using the Finite Element ToolKit (FETK), an open source finite element software package. We finish the paper with a sequence of numerical experiments illustrating the effectiveness of a number of BPX and stabilized HB variants for several examples requiring local refinement.

math.NA↗

arXiv2010

A Nonlinear Elasticity Model of Macromolecular Conformational Change Induced by Electrostatic Forces

In this paper we propose a nonlinear elasticity model of macromolecular conformational change (deformation) induced by electrostatic forces generated by an implicit solvation model. The Poisson-Boltzmann equation for the electrostatic potential is analyzed in a domain varying with the elastic deformation of molecules, and a new continuous model of the electrostatic forces is developed to ensure solvability of the nonlinear elasticity equations. We derive the estimates of electrostatic forces corresponding to four types of perturbations to an electrostatic potential field, and establish the existance of an equilibrium configuration using a fixed-point argument, under the assumption that the change in the ionic strength and charges due to the additional molecules causing the deformation are sufficiently small. The results are valid for elastic models with arbitrarily complex dielectric interfaces and cavities, and can be generalized to large elastic deformation caused by high ionic strength, large charges, and strong external fields by using continuation methods.

math.AP↗

arXiv2010

Local Convergence of Adaptive Methods for Nonlinear Partial Differential Equations

In this article we develop convergence theory for a general class of adaptive approximation algorithms for abstract nonlinear operator equations on Banach spaces, and use the theory to obtain convergence results for practical adaptive finite element methods (AFEM) applied to several classes of nonlinear elliptic equations. In the first part of the paper, we develop a weak-* convergence framework for nonlinear operators, whose Gateaux derivatives are locally Lipschitz and satisfy a local inf-sup condition. The framework can be viewed as extending the recent convergence results for linear problems of Morin, Siebert and Veeser to a general nonlinear setting. We formulate an abstract adaptive approximation algorithm for nonlinear operator equations in Banach spaces with local structure. The weak-* convergence framework is then applied to this class of abstract locally adaptive algorithms, giving a general convergence result. The convergence result is then applied to a standard AFEM algorithm in the case of several semilinear and quasi-linear scalar elliptic equations and elliptic systems, including: a semilinear problem with subcritical nonlinearity, the steady Navier-Stokes equations, and a quasilinear problem with nonlinear diffusion. This yields several new AFEM convergence results for these nonlinear problems. In the second part of the paper we develop a second abstract convergence framework based on strong contraction, extending the recent contraction results for linear problems of Cascon, Kreuzer, Nochetto, and Siebert and of Mekchay and Nochetto to abstract nonlinear problems. The contraction result is then applied to a standard AFEM algorithm for semilinear problems with sub- and super-critical nonlinearities and for the Hamiltonian constraint in general relativity.

math.NA↗

arXiv2009

Analysis of a General Family of Regularized Navier-Stokes and MHD Models

We consider a general family of regularized Navier-Stokes and Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian manifolds with or without boundary, with n greater than or equal to 2. This family captures most of the specific regularized models that have been proposed and analyzed in the literature, including the Navier-Stokes equations, the Navier-Stokes-alpha model, the Leray-alpha model, the Modified Leray-alpha model, the Simplified Bardina model, the Navier-Stokes-Voight model, the Navier-Stokes-alpha-like models, and certain MHD models, in addition to representing a larger 3-parameter family of models not previously analyzed. We give a unified analysis of the entire three-parameter family using only abstract mapping properties of the principle dissipation and smoothing operators, and then use specific parameterizations to obtain the sharpest results. We first establish existence and regularity results, and under appropriate assumptions show uniqueness and stability. We then establish results for singular perturbations, including the inviscid and alpha limits. Next we show existence of a global attractor for the general model, and give estimates for its dimension. We finish by establishing some results on determining operators for subfamilies of dissipative and non-dissipative models. In addition to establishing a number of results for all models in this general family, the framework recovers most of the previous results on existence, regularity, uniqueness, stability, attractor existence and dimension, and determining operators for well-known members of this family.

math.AP↗

arXiv2009

The Navier-Stokes-Voight Model for Image Inpainting

In 2001, Bertalmio et. al. drew an analogy between the image intensity function for the image inpainting problem and the stream function in a two-dimensional (2D) incompressible fluid. An approximate solution to the inpainting problem is obtained by numerically approximating the steady state solution of the 2D NSE vorticity transport equation, and simultaneously solving the Poisson problem between the vorticity and stream function, in the region to be inpainted. This elegant approach allows one to produce an approximate solution to the image inpainting problem by using techniques from computational fluid dynamics. Recently, the three-dimensional (3D) Navier-Stokes-Voight (NSV) model of viscoelastic fluid, was suggested by Cao et. al. as an inviscid regularization to the 3D Navier-Stokes equations (NSE). The NSV model is shown to be globally well-posed and has a finite-dimensional global attractor, making it an attractive sub-grid scale turbulence model for purposes of numerical simulation. In this paper we investigate the use of the 2D NSV model for use in algorithms for the inpainting problem. We also present some new theoretical results based on energy methods comparing the sufficient conditions for stability of the discretization scheme for the two model equations.

math.NA↗

arXiv2009

Solving the Einstein constraint equations on multi-block triangulations using finite element methods

In order to generate initial data for nonlinear relativistic simulations, one needs to solve the Einstein constraints, which can be cast into a coupled set of nonlinear elliptic equations. Here we present an approach for solving these equations on three-dimensional multi-block domains using finite element methods. We illustrate our approach on a simple example of Brill wave initial data, with the constraints reducing to a single linear elliptic equation for the conformal factor $ψ$. We use quadratic Lagrange elements on semi-structured simplicial meshes, obtained by triangulation of multi-block grids. In the case of uniform refinement the scheme is superconvergent at most mesh vertices, due to local symmetry of the finite element basis with respect to local spatial inversions. We show that in the superconvergent case subsequent unstructured mesh refinements do not improve the quality of our initial data. As proof of concept that this approach is feasible for generating multi-block initial data in three dimensions, after constructing the initial data we evolve them in time using a high order finite-differencing multi-block approach and extract the gravitational waves from the numerical solution.

gr-qc↗

arXiv2009

Generating Initial Data in General Relativity using Adaptive Finite Element Methods

The conformal formulation of the Einstein constraint equations is first reviewed, and we then consider the design, analysis, and implementation of adaptive multilevel finite element-type numerical methods for the resulting coupled nonlinear elliptic system. We derive weak formulations of the coupled constraints, and review some new developments in the solution theory for the constraints in the cases of constant mean extrinsic curvature (CMC) data, near-CMC data, and arbitrarily prescribed mean extrinsic curvature data. We then outline some recent results on a priori and a posteriori error estimates for a broad class of Galerkin-type approximation methods for this system which includes techniques such as finite element, wavelet, and spectral methods. We then use these estimates to construct an adaptive finite element method (AFEM) for solving this system numerically, and outline some new convergence and optimality results. We then describe in some detail an implementation of the methods using the FETK software package, which is an adaptive multilevel finite element code designed to solve nonlinear elliptic and parabolic systems on Riemannian manifolds. We finish by describing a simplex mesh generation algorithm for compact binary objects, and then look at a detailed example showing the use of FETK for numerical solution of the constraints.

gr-qc↗

arXiv2008

Rough solutions of the Einstein constraints on closed manifolds without near-CMC conditions

We consider the conformal decomposition of Einstein's constraint equations introduced by Lichnerowicz and York, on a closed manifold. We establish existence of non-CMC weak solutions using a combination of a priori estimates for the individual Hamiltonian and momentum constraints, barrier constructions for the Hamiltonian constraint, and topological fixed-point arguments. An important new feature of these results is the absense of the near-CMC assumption when the rescaled background metric is in the positive Yamabe class, if the freely specifiable part of the data given by the matter fields (if present) and the traceless-transverse part of the rescaled extrinsic curvature are taken to be sufficiently small. In this case, the mean extrinsic curvature can be taken to be an arbitrary smooth function without restrictions on the size of its spatial derivatives, giving what are apparently the first non-CMC existence results without the near-CMC assumption. Standard bootstrapping arguments to increase the regularity of the conformal factor are blocked by the use of a weak background metric. In the CMC case, we recover Maxwell's rough solution results as a special case. Our results extend the 1996 non-CMC result of Isenberg and Moncrief in three ways: (1) the near-CMC assumption is removed in the case of the positive Yamabe class; (2) regularity is extended down to the maximum allowed by the background metric and the matter; and (3) the result holds for all three Yamabe classes. This last extension was also accomplished recently by Allen, Clausen and Isenberg, although their result is restricted to the near-CMC case and to smoother background metrics and data.

gr-qc↗

arXiv2004

Optimal Constraint Projection for Hyperbolic Evolution Systems

Techniques are developed for projecting the solutions of symmetric hyperbolic evolution systems onto the constraint submanifold (the constraint-satisfying subset of the dynamical field space). These optimal projections map a field configuration to the ``nearest'' configuration in the constraint submanifold, where distances between configurations are measured with the natural metric on the space of dynamical fields. The construction and use of these projections is illustrated for a new representation of the scalar field equation that exhibits both bulk and boundary generated constraint violations. Numerical simulations on a black-hole background show that bulk constraint violations cannot be controlled by constraint-preserving boundary conditions alone, but are effectively controlled by constraint projection. Simulations also show that constraint violations entering through boundaries cannot be controlled by constraint projection alone, but are controlled by constraint-preserving boundary conditions. Numerical solutions to the pathological scalar field system are shown to converge to solutions of a standard representation of the scalar field equation when constraint projection and constraint-preserving boundary conditions are used together.

gr-qc↗