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53
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 50 (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 49 (6 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 Multilevel Methods in Three Space Dimensions
 Int. J. Numer. Methods Eng
, 1993
"... this paper to collect wellknown results on 3D mesh refinement ..."
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Cited by 42 (6 self)
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this paper to collect wellknown results on 3D mesh refinement
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 41 (24 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.
Optimality of a standard adaptive finite element method
"... In this paper, an adaptive ¯nite element method is constructed
for solving elliptic equations that has optimal computational complexity.
Whenever for some s > 0, the solution can be approximated to accuracy
O(n¡s) in energy norm by a continuous piecewise linear function on some
partition with n tria ..."
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Cited by 40 (3 self)
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In this paper, an adaptive ¯nite element method is constructed
for solving elliptic equations that has optimal computational complexity.
Whenever for some s > 0, the solution can be approximated to accuracy
O(n¡s) in energy norm by a continuous piecewise linear function on some
partition with n triangles, and one knows how to approximate the righthand
side in the dual norm with the same rate with piecewise constants, then
the adaptive method produces approximations that converge with this rate,
taking a number of operations that is of the order of the number of triangles
in the output partition. The method is similar in spirit to that from [SINUM,
38 (2000), pp.466{488] by Morin, Nochetto, and Siebert, and so in particular
it does not rely on a recurrent coarsening of the partitions. Although the
Poisson equation in two dimensions with piecewise linear approximation is
considered, it can be expected that the results generalize in several respects.
Hierarchical Bases
 In ICIAM91. SIAM
, 1992
"... . The solution of the large linear systems arising from finite element discretizations of elliptic boundary value problems is a basic task in numerical analysis. For uniformly refined grids, multigrid methods are well established and very efficient methods to solve these problems. For adaptively gen ..."
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Cited by 30 (2 self)
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. The solution of the large linear systems arising from finite element discretizations of elliptic boundary value problems is a basic task in numerical analysis. For uniformly refined grids, multigrid methods are well established and very efficient methods to solve these problems. For adaptively generated, strongly nonuniform grids, the situation is more complicated, both with regard to convergence properties and with regard to the computational complexity of the single iteration step. The paper discusses a surprisingly simple approach which is very well suited to nonuniform grids, the hierarchical decomposition of finite element spaces. The method is based on an old idea from the theory of real functions, which is often used to produce fractal curves and surfaces. It is closely related to recent L 2 like decompositions. Key words. hierarchical bases, fast iterative solvers, finite elements AMS(MOS) subject classifications. 65N22, 65N30, 65N50, 65N55, 26A27 1. Introduction. Toda...
Multiresolutional Parallel Isosurface Extraction based on Tetrahedral Bisection
 In Proc. Symp. Volume Visualization
, 1999
"... Nowadays, multiresolution visualization methods become an indispensable ingredient of real time interactive post processing. We will here present an efficient approach for tetrahedral grids recursively generated by bisection, which is based on a more general method for arbitrary nested grids. It esp ..."
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Cited by 28 (6 self)
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Nowadays, multiresolution visualization methods become an indispensable ingredient of real time interactive post processing. We will here present an efficient approach for tetrahedral grids recursively generated by bisection, which is based on a more general method for arbitrary nested grids. It especially applies to regular grids, the hexahedra of which are procedurally subdivided into tetrahedra. Besides different types of error indicators, we especially focus on improving the algorithm's performance and reducing the memory requirements. Furthermore, parallelization combined with an appropriate load balancing on multiprocessor workstations is discussed. 1 Introduction A variety of multiresolution visualization methods has been designed to serve as tools for interactive visualization of large data sets. The local resolution of the generated visual objects, such as isosurfaces, is thereby steered by error indicators which measure the error due to a locally coarser approximation of the...
The completion of locally refined simplicial partitions created by bisection
, 2006
"... Abstract. Recently, in [Found. Comput. Math., 7(2) (2007), 245–269], we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations, which is essentially the method from [SINUM, 38 (2000), 466–488] by Morin, Nochetto, and Sieb ..."
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Cited by 26 (2 self)
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Abstract. Recently, in [Found. Comput. Math., 7(2) (2007), 245–269], we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations, which is essentially the method from [SINUM, 38 (2000), 466–488] by Morin, Nochetto, and Siebert, converges with the optimal rate.The number of triangles N in the output partition of such a method is generally larger than the number M of triangles that in all intermediate partitions have been marked for bisection, because additional bisections are needed to retain conforming meshes.A key ingredient to our proof was a result from [Numer. Math., 97(2004), 219–268] by Binev, Dahmen and DeVore saying that N −N0 ≤ CM for some absolute constant C, where N0 is the number of triangles from the initial partition that have never been bisected. In this paper, we extend this result to bisection algorithms of nsimplices, with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions. 1.
A posteriori error estimates for vertex centered finite volume approximations of convectiondiffusionreaction equations
 M2AN Math. Model. Numer. Anal
, 2000
"... This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convectiondiffusionreaction equation c t +r (uf(c)) r (Drc)+c = 0. The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the L¹ ..."
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Cited by 24 (7 self)
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This paper is devoted to the study of a posteriori error estimates for the scalar nonlinear convectiondiffusionreaction equation c t +r (uf(c)) r (Drc)+c = 0. The estimates for the error between the exact solution and an upwind finite volume approximation to the solution are derived in the L¹norm, independent of the diffusion parameter D. The resulting a posteriori error estimate is used to define an grid adaptive solution algorithm for the finite volume scheme. Finally numerical
A posteriori error estimation and adaptivity for degenerate parabolic problems
 Math. Comp
, 2000
"... Abstract. Two explicit error representation formulas are derived for degenerate parabolic PDEs, which are based on evaluating a parabolic residual in negative norms. The resulting upper bounds are valid for any numerical method, and rely on regularity properties of solutions of a dual parabolic prob ..."
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Cited by 24 (10 self)
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Abstract. Two explicit error representation formulas are derived for degenerate parabolic PDEs, which are based on evaluating a parabolic residual in negative norms. The resulting upper bounds are valid for any numerical method, and rely on regularity properties of solutions of a dual parabolic problem in nondivergence form with vanishing diffusion coefficient. They are applied to a practical spacetime discretization consisting of C 0 piecewise linear finite elements over highly graded unstructured meshes, and backward finite differences with varying timesteps. Two rigorous a posteriori error estimates are derived for this scheme, and used in designing an efficient adaptive algorithm, which equidistributes space and time discretization errors via refinement/coarsening. A simulation finally compares the behavior of the rigorous a posteriori error estimators with a heuristic approach, and hints at the potentials and reliability of the proposed method. 1.