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118
UG  A Flexible Software Toolbox For Solving Partial Differential Equations
 COMPUTING AND VISUALIZATION IN SCIENCE
, 1997
"... Over the past two decades, some very efficient techniques for the numerical solution of partial differential equations have been developed. We are especially interested in adaptive local grid refinement on unstructured meshes, multigrid solvers and parallelization techniques. Up to now, these innova ..."
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Cited by 109 (21 self)
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Over the past two decades, some very efficient techniques for the numerical solution of partial differential equations have been developed. We are especially interested in adaptive local grid refinement on unstructured meshes, multigrid solvers and parallelization techniques. Up to now, these innovative techniques have been implemented mostly in university research codes and only very few commercial codes use them. There are two reasons for this. Firstly, the multigrid solution and adaptive refinement for many engineering applications are still a topic of active research and cannot be considered to be mature enough for routine application. Secondly, the implementation of all these techniques in a code with sufficient generality requires a lot of time and knowhow in different fields. UG (abbreviation for Unstructured Grids) has been designed to overcome these problems. It provides very general tools for the generation and manipulation of unstructured meshes in two and three space dime...
Concepts of an adaptive hierarchical finite element code
 IMPACT COMPUT. SCI. ENGRG
, 1989
"... The paper presents the mathematical concepts underlying the new adaptive finite element code KASKADE, which, in its present form, applies to linear scalar secondorder 2D elliptic problems on general domains. Starting point for the new development is the recent work on hierarchical finite element b ..."
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Cited by 99 (10 self)
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The paper presents the mathematical concepts underlying the new adaptive finite element code KASKADE, which, in its present form, applies to linear scalar secondorder 2D elliptic problems on general domains. Starting point for the new development is the recent work on hierarchical finite element bases due to Yserentant (1986). It is shown that this approach permits a flexible balance between iterative solver, local error estimator, and local mesh refinement device which are the main components of an adaptive PDE code. Without use of standard multigrid techniques, the same kind of computational complexity is achieved independent of any uniformity restrictions on the applied meshes. In addition, the method is extremely simple and all computations are purely local making the method particularly attractive in view of parallel computing. The algorithmic approach is
The Adaptive Multilevel Finite Element Solution of the PoissonBoltzmann Equation on Massively Parallel Computers
 J. COMPUT. CHEM
, 2000
"... Using new methods for the parallel solution of elliptic partial differential 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 soluti ..."
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Cited by 90 (17 self)
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Using new methods for the parallel solution of elliptic partial differential 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 PoissonBoltzmann 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 of1800 e. PoissonBoltzmann calculations are performed for several processor configurations and the algorithm shows excellent parallel scaling.
Hierarchical Bases and the Finite Element Method
, 1997
"... CONTENTS 1 Introduction 1 2 Preliminaries 3 3 Fundamental TwoLevel Estimates 7 4 A Posteriori Error Estimates 16 5 TwoLevel Iterative Methods 23 6 Multilevel Cauchy Inequalities 30 7 Multilevel Iterative Methods 34 References 41 1. Introduction In this work we present a brief introduction to hie ..."
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Cited by 80 (4 self)
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CONTENTS 1 Introduction 1 2 Preliminaries 3 3 Fundamental TwoLevel Estimates 7 4 A Posteriori Error Estimates 16 5 TwoLevel Iterative Methods 23 6 Multilevel Cauchy Inequalities 30 7 Multilevel Iterative Methods 34 References 41 1. Introduction In this work we present a brief introduction to hierarchical bases, and the important part they play in contemporary finite element calculations. In particular, we examine their role in a posteriori error estimation, and in the Department of Mathematics, University of California at San Diego, La Jolla, CA 92093. The work of this author was supported by the Office of Naval Research under contract N0001489J1440. 2 Randolph E. Bank formulation of iterative methods for solving the large sparse sets of linear equations arising from the finite element discretization. Our goal is that the development should be largely selfcontained, but at the same time accessible and interest
A Posteriori Error Estimates Based On Hierarchical Bases
 SIAM JOURNAL ON NUMERICAL ANALYSIS
, 1993
"... The authors present an analysis of an a posteriori error estimator based on the use of hierarchical basis functions. The authors analyze nonlinear, nonselfadjoint and indefinite problems as well as the selfadjoint, positivedefinite case. Because both the analysis and the estimator itself are quite ..."
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Cited by 76 (4 self)
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The authors present an analysis of an a posteriori error estimator based on the use of hierarchical basis functions. The authors analyze nonlinear, nonselfadjoint and indefinite problems as well as the selfadjoint, positivedefinite case. Because both the analysis and the estimator itself are quite simple, it is easy to see how various approximations affect the quality of the estimator. As examples, the authors apply the theory to some scalar elliptic equations and the Stokes system of equations.
A Class Of Iterative Methods For Solving Saddle Point Problems
 Numer. Math
, 1990
"... . We consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points. We are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations. We present an iterative method involving two ..."
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Cited by 75 (3 self)
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. We consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points. We are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations. We present an iterative method involving two levels of iteration, similar in some respects to the Uzawa algorithm. We relate the rates of convergence of the outer and inner iterations, proving that, under natural hypotheses, the outer iteration achieves the rate of convergence of the inner iteration. The technique is applied to finite element approximations of the Stokes equations. Key words. Saddle point problems, iterative solvers, mixed finite element methods. AMS subject classifications. 65F10, 65N20, 65N30. 1. Introduction. In this paper, we consider the solution of the system of linear equations A B t B 0 x y = f g (1.1) Here A is a symmetric, positive definite n \Theta n matrix, and B is an m \Theta n matrix....
An Adaptive Multilevel Approach to Parabolic Equations in Two Space Dimensions
, 1991
"... A new adaptive multilevel approach, for linear parabolic partial differential equations is presented, which is able to handle complicated space geometries, discontinuous coefficients, inconsistent initial data. Discretization in time first (Rothe's method) with order and stepsize control is per ..."
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Cited by 57 (6 self)
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A new adaptive multilevel approach, for linear parabolic partial differential equations is presented, which is able to handle complicated space geometries, discontinuous coefficients, inconsistent initial data. Discretization in time first (Rothe's method) with order and stepsize control is perturbed by an adaptive finite element discretization of the elliptic subproblems, whose errors are controlled independently. Thus the high standards of solving adaptively ordinary differential equations and elliptic boundary value problems are combined. A theory of time discretization in Hilbert space is developed which yields to an optimal variable order method based on a multiplicative error correction. The problem of an efficient solution of the singularly perturbed elliptic subproblems and the problem of error estimation for them can be uniquely solved within the framework of preconditioning. A multilevel nodal basis preconditioner is derived, which allows
Adaptive numerical treatment of elliptic systems on manifolds
 Advances in Computational Mathematics, 15(1):139
, 2001
"... 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 ..."
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Cited by 56 (25 self)
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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 3manifolds 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 3manifolds. 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 unitybased 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 4component covariant elliptic system on a Riemannian 3manifold which arises in general relativity. A number of operator properties and solvability results recently established are first summarized, making possible two quasioptimal 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.