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174
The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources
 SIAM J. Num. Anal
, 1994
"... Abstract. The authors develop finite difference methods for elliptic equations of the form V. ((x)Vu(x)) + (x)u(x) f(x) in a region in one or two space dimensions. It is assumed that gt is a simple region (e.g., a rectangle) and that a uniform rectangular grid is used. The situation is studied in wh ..."
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Cited by 270 (31 self)
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Abstract. The authors develop finite difference methods for elliptic equations of the form V. ((x)Vu(x)) + (x)u(x) f(x) in a region in one or two space dimensions. It is assumed that gt is a simple region (e.g., a rectangle) and that a uniform rectangular grid is used. The situation is studied in which there is an irregular surface F of codimension contained in fl across which, a, and f may be discontinuous, and along which the source f may have a delta function singularity. As a result, derivatives of the solution u may be discontinuous across F. The specification of a jump discontinuity in u itself across F is allowed. It is shown that it is possible to modify the standard centered difference approximation to maintain second order accuracy on the uniform grid even when F is not aligned with the grid. This approach is also compared with a discrete delta function approach to handling singular sources, as used in Peskin’s immersed boundary method. Key words, elliptic equation, finite difference methods, irregular domain, interface, discontinuous coefficients, singular source term, delta functions AMS subject classifications. 65N06, 65N50 1. Introduction. Consider
deal.II – a general purpose object oriented finite element library
 ACM TRANS. MATH. SOFTW
"... An overview of the software design and data abstraction decisions chosen for deal.II, a general purpose finite element library written in C++, is given. The library uses advanced objectoriented and data encapsulation techniques to break finite element implementations into smaller blocks that can be ..."
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Cited by 105 (28 self)
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An overview of the software design and data abstraction decisions chosen for deal.II, a general purpose finite element library written in C++, is given. The library uses advanced objectoriented and data encapsulation techniques to break finite element implementations into smaller blocks that can be arranged to fit users requirements. Through this approach, deal.II supports a large number of different applications covering a wide range of scientific areas, programming methodologies, and applicationspecific algorithms, without imposing a rigid framework into which they have to fit. A judicious use of programming techniques allows to avoid the computational costs frequently associated with abstract objectoriented class libraries. The paper presents a detailed description of the abstractions chosen for defining geometric information of meshes and the handling of degrees of freedom associated with finite element spaces, as well as of linear algebra, input/output capabilities and of interfaces to other software, such as visualization tools. Finally, some results obtained with applications built atop deal.II are shown to demonstrate the powerful capabilities of this toolbox.
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 89 (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.
High contrast impedance tomography
 INVERSE PROBLEMS
, 1996
"... We introduce an output leastsquares method for impedance tomography problems that have regions of high conductivity surrounded by regions of lower conductivity. The high conductivity is modeled on network approximation results from an asymptotic analysis and its recovery is based on this model. The ..."
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Cited by 86 (7 self)
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We introduce an output leastsquares method for impedance tomography problems that have regions of high conductivity surrounded by regions of lower conductivity. The high conductivity is modeled on network approximation results from an asymptotic analysis and its recovery is based on this model. The smoothly varying part of the conductivity is recovered by a linearization process as is usual. We present the results of several numerical experiments that illustrate
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 78 (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.
Quality local refinement of tetrahedral meshes based on bisection
 SIAM J. Sci. Comput
, 1995
"... Abstract. Let T be a tetrahedral mesh. We present a 3D local refinement algorithm for T which is mainly based on an 8subtetrahedron subdivision procedure, and discuss the quality of refined meshes generated by the algorithm. It is proved that any tetrahedron T ∈T produces a finite number of classe ..."
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Cited by 72 (1 self)
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Abstract. Let T be a tetrahedral mesh. We present a 3D local refinement algorithm for T which is mainly based on an 8subtetrahedron subdivision procedure, and discuss the quality of refined meshes generated by the algorithm. It is proved that any tetrahedron T ∈T produces a finite number of classes of similar tetrahedra, independent of the number of refinement levels. Furthermore, η(Tn i) ≥ cη(T), where T ∈T,cis a positive constant independent of T and the number of refinement levels, Tn i is any refined tetrahedron of T, andηisa tetrahedron shape measure. It is also proved that local refinements on tetrahedra can be smoothly extended to their neighbors to maintain a conforming mesh. Experimental results show that the ratio of the number of tetrahedra actually refined to the number of tetrahedra chosen for refinement is bounded above by a small constant. 1.
LOCALLY ADAPTED TETRAHEDRAL MESHES USING BISECTION
, 2000
"... We present an algorithm for the construction of locally adapted conformal tetrahedral meshes. The algorithm is based on bisection of tetrahedra. A new data structure is introduced, which simplifies both the selection of the refinement edge of a tetrahedron and the recursive refinement to conformity ..."
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Cited by 68 (1 self)
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We present an algorithm for the construction of locally adapted conformal tetrahedral meshes. The algorithm is based on bisection of tetrahedra. A new data structure is introduced, which simplifies both the selection of the refinement edge of a tetrahedron and the recursive refinement to conformity of a mesh once some tetrahedra have been bisected. We prove that repeated application of the algorithm leads to only finitely many tetrahedral shapes up to similarity, and we bound the amount of additional refinement that is needed to achieve conformity. Numerical examples of the effectiveness of the algorithm are presented.
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 61 (8 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 57 (26 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.