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65
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 65 (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.
Mesh Generation
 Handbook of Computational Geometry. Elsevier Science
, 2000
"... this article, we emphasize practical issues; an earlier survey by Bern and Eppstein [24] emphasized theoretical results. Although there is inevitably some overlap between these two surveys, we intend them to be complementary. ..."
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Cited by 55 (8 self)
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this article, we emphasize practical issues; an earlier survey by Bern and Eppstein [24] emphasized theoretical results. Although there is inevitably some overlap between these two surveys, we intend them to be complementary.
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 54 (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.
WellSpaced Points for Numerical Methods
, 1997
"... mesh generation, mesh coarsening, multigrid Abstract A numerical method for the solution of a partial differential equation (PDE) requires the following steps: (1) discretizing the domain (mesh generation); (2) using an approximation method and the mesh to transform the problem into a linear system; ..."
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Cited by 51 (2 self)
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mesh generation, mesh coarsening, multigrid Abstract A numerical method for the solution of a partial differential equation (PDE) requires the following steps: (1) discretizing the domain (mesh generation); (2) using an approximation method and the mesh to transform the problem into a linear system; (3) solving the linear system. The approximation error and convergence of the numerical method depend on the geometric quality of the mesh, which in turn depends on the size and shape of its elements. For example, the shape quality of a triangular mesh is measured by its element's aspect ratio. In this work, we shift the focus to the geometric properties of the nodes, rather than the elements, of well shaped meshes. We introduce the concept of wellspaced points and their spacing functions, and show that these enable the development of simple and efficient algorithms for the different stages of the numerical solution of PDEs. We first apply wellspaced point sets and their accompanying technology to mesh coarsening, a crucial step in the multigrid solution of a PDE. A good aspectratio coarsening sequence of an unstructured mesh M0 is a sequence of good aspectratio meshes M1; : : : ; Mk such that Mi is an approximation of Mi\Gamma 1 containing fewer nodes and elements. We present a new approach to coarsening that guarantees the sequence is also of optimal size and width up to a constant factor the first coarsening method that provides these guarantees. We also present experimental results, based on an implementation of our approach, that substantiate the theoretical claims.
A 3D unstructured mesh adaption algorithm for time dependent shock dominated problems
, 1995
"... In this paper we present a tetrahedral based, hrefinement type, algorithm for the solution of problems in 3D gas dynamics using unstructured mesh adaptation. The mesh adaptation algorithm is coupled to a cell centred, Riemann Problem based, finite volume scheme of the MUSCL type, employing an appro ..."
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Cited by 41 (9 self)
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In this paper we present a tetrahedral based, hrefinement type, algorithm for the solution of problems in 3D gas dynamics using unstructured mesh adaptation. The mesh adaptation algorithm is coupled to a cell centred, Riemann Problem based, finite volume scheme of the MUSCL type, employing an approximate Riemann solver. The adaptive scheme is then used to compute the diffraction of shock waves around a box section corner for subsonic and supersonic post shock flow. In the subsonic case preliminary measurements of vortex filament speed and vortical mach number are in broad quantitative agreement with known theoretical results. 1. INTRODUCTION The numerical investigation of phenomena associated with shock wave propagation through the use of conservative shockcapturing, highresolution, RiemannProblem based numerical methods for hyperbolic conservation laws, has generated great interest within the fluid dynamics community over recent years. 21 A number of high quality two dimensiona...
Simplicial Grid Refinement: On Freudenthal's Algorithm and the Optimal Number of Congruence Classes
 NUMER. MATH
, 1998
"... In the present paper we investigate Freudenthal's simplex refinement algorithm which can be considered to be the canonical generalization of Bank's well known red refinement strategy for triangles. Freudenthal's algorithm subdivides any given (n)simplex into 2 n subsimplices, in su ..."
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Cited by 31 (0 self)
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In the present paper we investigate Freudenthal's simplex refinement algorithm which can be considered to be the canonical generalization of Bank's well known red refinement strategy for triangles. Freudenthal's algorithm subdivides any given (n)simplex into 2 n subsimplices, in such a way that recursive application results in a stable hierarchy of consistent triangulations. Our investigations concentrate in particular on the number of congruence classes generated by recursive refinements. After presentation of the method and the basic ideas behind it, we will show that Freudenthal's algorithm produces at most n!=2 congruence classes for any initial (n)simplex, no matter how many subsequent refinements are performed. Moreover, we will show that this number is optimal in the sense that recursive application of any affine invariant refinement strategy with 2 n sons per element results in at least n!=2 congruence classes for almost all (n)simplices.
An Odyssey Into Local Refinement And Multilevel Preconditioning I: Optimality Of . . .
 SIAM J. NUMER. ANAL
, 2002
"... In this article, we examine the BramblePasciakXu (BPX) preconditioner in the setting of local 2D and 3D mesh refinement. While the available optimality results for the BPX preconditioner have been constructed primarily in the setting of uniformly refined meshes, a notable exception is the 2D resul ..."
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Cited by 27 (14 self)
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In this article, we examine the BramblePasciakXu (BPX) preconditioner in the setting of local 2D and 3D mesh refinement. While the available optimality results for the BPX preconditioner have been constructed primarily in the setting of uniformly refined meshes, a notable exception is the 2D result due to Dahmen and Kunoth, which established BPX optimality on meshes produced by a restricted class of local 2D redgreen refinement. The purpose of this article is to extend the original 2D DahmenKunoth result to several additional types of local 2D and 3D redgreen (conforming) and red (nonconforming) refinement procedures. The extensions are accomplished through a 3D extension of the 2D framework in the original DahmenKunoth work, by which the question of optimality is reduced to establishing that locally enriched finite element subspaces allow for the construction of a scaled basis which is formally Riesz stable. This construction in turn rests entirely on establishing a number of geometrical properties between neighboring simplices produced by the local refinement algorithms. These properties are then used to build Rieszstable scaled bases for use in the BPX optimality framework. Since the theoretical framework supports arbitrary spatial dimension d 1, we indicate clearly which geometrical properties, established here for several 2D and 3D local refinement procedures, must be reestablished to show BPX optimality for spatial dimension 4. Finally, we also present a simple alternative optimality proof of the BPX preconditioner on quasiuniform meshes in two and three spatial dimensions, through the use of Kfunctionals and H stability of L 2 projection for s 1. The proof techniques we use are quite general; in particular, the results require no smoothnes...
Natural Hierarchical Refinement for Finite Element Methods
 International Journal for Numerical Methods in Engineering
, 2001
"... Introduction Adaptive finite element computations rely on adjustments of th spatial resolution of th domain discretization to deliverh igh er accuracywh ere it is neededWh enth e domain is discretized into a finite element mesh a possible option, albeit somewh at expensive and in some cases complex ..."
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Cited by 26 (8 self)
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Introduction Adaptive finite element computations rely on adjustments of th spatial resolution of th domain discretization to deliverh igh er accuracywh ere it is neededWh enth e domain is discretized into a finite element mesh a possible option, albeit somewh at expensive and in some cases complex, is to create a new mesh with th desired resolution, i e remeshing Anoth0 alternative is to adjustthdensityofth mesh by performing local refinement (or coarsening) of th existing mesh soth t in some regions finite elements are split to decreaseth eir "size", inoth er regionsth ey are merged to reduce th e resolution Both ch oices, remesh ing and refinement,h ave th eir advantages and disadvantages We are not going to argue for one or th oth r option Rath r, we assume th t refinement h d been adopted asth meth d of ch ice Wh t are th desirable properties of a mesh refinement algorith0 It sh uld certainly be e#cient in th at itsh ould not becom
Optimality of multilevel preconditioners for local mesh refinement in three dimensions
 SIAM J. Numer. Anal
"... Abstract. 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 e ..."
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Cited by 20 (10 self)
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Abstract. 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 redgreen result due to Dahmen and Kunoth. The purpose of this article is to extend this original 2D optimality result to the local 3D redgreen refinement procedure introduced by Bornemann, Erdmann, and Kornhuber, and then to use this result to extend the WHB optimality results from the quasiuniform setting to local 2D and 3D redgreen 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. This construction turns out to rest not only on the shape regularity of the refined elements, but also critically on a number of geometrical properties we establish between neighboring simplices produced by the Bornemann–Erdmann–Kornhuber (BEK) refinement procedure. 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 H 1stability of the L2projection. The existing a posteriori approaches in the literature dictate a reconstruction of the mesh if such conditions cannot be satisfied. The theoretical framework employed supports arbitrary spatial dimension d ≥ 1 and requires no coefficient smoothness assumptions beyond those required for wellposedness in H 1.
Spacetime Meshing with Adaptive Refinement and Coarsening
 SCG'04
, 2004
"... We propose a new algorithm for constructing finiteelement meshes suitable for spacetime discontinuous Galerkin solutions of linear hyperbolic PDEs. Given a triangular mesh of some planar domain# and a target time value T , our method constructs a tetrahedral mesh of the spacetime domain [0, T] i ..."
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Cited by 19 (10 self)
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We propose a new algorithm for constructing finiteelement meshes suitable for spacetime discontinuous Galerkin solutions of linear hyperbolic PDEs. Given a triangular mesh of some planar domain# and a target time value T , our method constructs a tetrahedral mesh of the spacetime domain [0, T] in constant running time per tetrahedron in IR using an advancing front method. Elements are added to the evolving mesh in small patches by moving a vertex of the front forward in time. Spacetime discontinuous Galerkin methods allow the numerical solution within each patch to be computed as soon as the patch is created. Our algorithm employs new mechanisms for adaptively coarsening and refining the front in response to a posteriori error estimates returned by the numerical code. A change in the front induces a corresponding refinement or coarsening of future elements in the spacetime mesh. Our algorithm adapts the duration of each element to the local quality, feature size, and degree of refinement of the underlying space mesh. We directly exploit the ability of discontinuous Galerkin methods to accommodate discontinuities in the solution fields across element boundaries.