this paper.

ABSTRACT. 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 developed over several years at Caltech and UC San Diego. 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.

Using new methods for the parallel solution of elliptic partial di#erential equations, the teraflops computing power of massively parallel computers can be leveraged to perform electrostatic calculations on large biological systems. This paper describes the adaptive multilevel finite element solution of the Poisson-Boltzmann equation for a microtubule on the NPACI IBM Blue Horizon supercomputer. The microtubule system is 40 nm in length and 24 nm in diameter, consists of roughly 600,000 atoms, and has a net charge of-1800 e. Poisson-Boltzmann calculations are performed for several processor configurations and the algorithm shows excellent parallel scaling.

Abstract. A computable error bound for mixed finite element methods is established in the model case of the Poisson–problem to control the error in the H(div,Ω) ×L 2 (Ω)–norm. The reliable and efficient a posteriori error estimate applies, e.g., to Raviart–Thomas, Brezzi-Douglas-Marini, and Brezzi-Douglas-Fortin-Marini elements. 1. Mixed method for the Poisson problem Mixed finite element methods are well-established in the numerical treatment of partial differential equations as regards a priori error estimates to guarantee convergence [BF]. In practical applications, a posteriori error control is at least of the same importance to guarantee a reliable approximation. Moreover, a posteriori error estimators indicate adaptive mesh-refinement criteria [EEHJ, V1] for an efficient computation. In this paper we establish an efficient and reliable error estimator for the model example in the mixed finite element methods: Given f ∈ L2 (Ω), the Poisson problem consists in finding a function u ∈ H1 0 (Ω) that satisfies (1.1) div(A∇u)+f =0 inΩ. Here, A ∈ L ∞ (Ω; R2×2) is symmetric and uniformly elliptic, Ω is a convex bounded domain in the plane with polygonal boundary Γ. The Lebesgue and Sobolev spaces (Ω) are defined as usual (e.g., as in [H, LM]). We assume below that L2 (Ω) and H1 0 (1.1) is H2 –regular which, according to Ω being convex, means certain regularity on A (A the unit matrix as for the Laplace equation is clearly sufficient). The mixed formulation is given by splitting (1.1) into two equations where u ∈ H1 0 (Ω) and p ∈ L2 (Ω) 2 are unknown and have to satisfy (1.2) div p + f =0 and p=A∇u in Ω. It is well-known that (1.2) has a solution (p, u) ∈ H(div,Ω)×L 2 (Ω), where, as usual, H(div,Ω): = {q ∈ L 2 (Ω) 2:divq∈L 2 (Ω)} is endowed with the norm given by

Abstract. The direct numerical solution of a non-convex variational problem (P) typically faces the difficulty of the finite element approximation of rapid oscillations. Although the oscillatory discrete minimisers are properly related to corresponding Young measures and describe real physical phenomena, they are costly and difficult to compute. In this work, we treat the scalar double-well problem by numerical solution of the relaxed problem (RP) leading to a (degenerate) convex minimisation problem. The problem (RP) has a minimiser u and a related stress field σ = DW ∗ ∗ (∇u) which is known to coincide with the stress field obtained by solving (P) in a generalised sense involving Young measures. If uh is a finite element solution, σh: = DW ∗ ∗ (∇uh) is the related discrete stress field. We prove a priori and a posteriori estimates for σ − σh in L 4/3 (Ω) and weaker weighted estimates for ∇u −∇uh. The a posteriori estimate indicates an adaptive scheme for automatic mesh refinements as illustrated in numerical experiments. 1.

Abstract. A number of new local and parallel discretization and adaptive finite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems. These algorithms are motivated by the observation that, for a solution to some elliptic problems, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. The theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for finite element solutions on general shape-regular grids. Some numerical experiments are also presented to support the theory. 1.

linear parabolic equations

Babuska and Yu constructed a posteriori estimates for finite element discretization errors of linear elliptic problems utilizing a dichotomy principal stating that the errors of odd-order approximations arise near element edges as mesh spacing decreases while those of even-order approximations arise in element interiors. We construct similar a posteriori estimates for the spatial errors of finite element method-of-lines solutions of linear parabolic partial differential equations on square-element meshes. Error estimates computed in this manner are proven to be asymptotically correct; thus, they converge in strain energy under mesh refinement at the same rate as the actual errors. 1. Introduction A posteriori estimates of discretization errors have been an integral part of adaptive finite element methods since their inception nearly twenty years ago and are used for elliptic 4;5;6;7;11;16;29;30 and parabolic problems. 3;14;15;17;20;21;24 Local contributions to global error estimat...

An ecient and reliable a posteriori error estimate is derived for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator. An adaptive algorithm with variable time-step sizes and space meshes is proposed and studied which, at each time step, delays the mesh coarsening until the nal iteration of the adaptive procedure, allowing only mesh and time-step size re nements before. It is proved that at each time step, the adaptive algorithm is able to reduce the error indicators (and thus the error) below any given tolerance within nite number of iteration steps. The key ingredient in the analysis is a new coarsening strategy. Numerical results are presented to show the competitive behavior of the proposed adaptive algorithm. 1.

ABSTRACT. A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson–Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson–Boltzmann equation is introduced as an auxiliary problem, making it possible to study the original nonlinear equation with delta distribution sources. A priori error estimates for the finite element approximation are obtained for the regularized Poisson–Boltzmann equation based on certain quasi-uniform grids in two and three dimensions. Adaptive finite element approximation through local refinement driven by an a posteriori error estimate is shown to converge. The Poisson–Boltzmann equation does not appear to have been previously studied in detail theoretically, and it is hoped that this paper will help provide molecular modelers with a better foundation for their analytical and computational work with the Poisson–Boltzmann equation. Note that this article apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear Poisson– Boltzmann equation with delta distribution sources, and it also introduces the first provably convergent adaptive method for the equation. This last result is currently one of only a handful of existing convergence results of this type for nonlinear problems.