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Anisotropic Voronoi Diagrams and GuaranteedQuality Anisotropic Mesh Generation
 in SCG ’03: Proceedings of the nineteenth annual symposium on Computational geometry
, 2003
"... We introduce anisotropic Voronoi diagrams, a generalization of multiplicatively weighted Voronoi diagrams suitable for generating guaranteedquality meshes of domains in which long, skinny triangles are required, and where the desired anisotropy varies over the domain. We discuss properties of aniso ..."
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Cited by 61 (2 self)
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We introduce anisotropic Voronoi diagrams, a generalization of multiplicatively weighted Voronoi diagrams suitable for generating guaranteedquality meshes of domains in which long, skinny triangles are required, and where the desired anisotropy varies over the domain. We discuss properties of anisotropic Voronoi diagrams of arbitrary dimensionalitymost notably circumstances in which a site can see its entire Voronoi cell. In two dimensions, the anisotropic Voronoi diagram dualizes to a triangulation under these same circumstances. We use these properties to develop an algorithm for anisotropic triangular mesh generation in which no triangle has an angle smaller than 20 # , as measured from the skewed perspective of any point in the triangle.
What Is a Good Linear Finite Element?  Interpolation, Conditioning, Anisotropy, and Quality Measures
 In Proc. of the 11th International Meshing Roundtable
, 2002
"... When a mesh of simplicial elements (triangles or tetrahedra) is used to form a piecewise linear approximation of a function, the accuracy of the approximation depends on the sizes and shapes of the elements. In finite element methods, the conditioning of the stiffness matrices also depends on the si ..."
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Cited by 59 (0 self)
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When a mesh of simplicial elements (triangles or tetrahedra) is used to form a piecewise linear approximation of a function, the accuracy of the approximation depends on the sizes and shapes of the elements. In finite element methods, the conditioning of the stiffness matrices also depends on the sizes and shapes of the elements. This article explains the mathematical connections between mesh geometry, interpolation errors, discretization errors, and stiffness matrix conditioning. These relationships are expressed by error bounds and element quality measures that determine the fitness of a triangle or tetrahedron for interpolation or for achieving low condition numbers. Unfortunately, the quality measures for these purposes do not fully agree with each other; for instance, small angles are bad for matrix conditioning but not for interpolation or discretization. The upper and lower bounds on interpolation error and element stiffness matrix conditioning given here are tighter than those usually seen in the literature, so the quality measures are likely to be unusually precise indicators of element fitness. Bounds are included for anisotropic cases, wherein long, thin elements perform better than equilateral ones. Surprisingly, there are circumstances wherein interpolation, conditioning, and discretization error are each best served by elements of different aspect ratios or orientations.
Piecewise Tensor Product Wavelet Bases by Extensions and Approximation Rates
, 2011
"... The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distrib ..."
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Cited by 46 (6 self)
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The consecutive numbering of the publications is determined by their chronological order. The aim of this preprint series is to make new research rapidly available for scientific discussion. Therefore, the responsibility for the contents is solely due to the authors. The publications will be distributed by the authors.
Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs. Report 201003, Seminar for Applied Mathematics
, 2010
"... Parametric partial differential equations are commonly used to model physical systems. They also arise when Wiener chaos expansions are used as an alternative to Monte Carlo when solving stochastic elliptic problems. This paper considers a model class of second order, linear, parametric, elliptic PD ..."
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Cited by 43 (10 self)
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Parametric partial differential equations are commonly used to model physical systems. They also arise when Wiener chaos expansions are used as an alternative to Monte Carlo when solving stochastic elliptic problems. This paper considers a model class of second order, linear, parametric, elliptic PDEs in a bounded domain D with coefficients depending on possibly countably many parameters. It shows that the dependence of the solution on the parameters in the diffusion coefficient is analytically smooth. This analyticity is then exploited to prove that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations can be simultaneously approximated by multivariate polynomials (in the parameters) with coefficients taking values in the Hilbert space V = H10 (D) of weak solutions of the elliptic problem with a controlled number of terms N. The convergence rate in terms of N does not depend on the number of parameters in V which may be countable, therefore breaking the curse of dimensionality. The discretization of the coefficients from a family of continuous, piecewise linear Finite Element functions in D is shown to yield finite dimensional approximations whose convergence rate in terms of the overall number Ndof of degrees of freedom is the minimum of the convergence rates afforded by the best Nterm sequence approximations in the parameter space and the rate of Finite Element approximations in D for a single instance of the parametric problem. 1
Aggressive Tetrahedral Mesh Improvement
 In Proc. of the 16th Int. Meshing Roundtable
, 2007
"... Summary. We present a tetrahedral mesh improvement schedule that usually creates meshes whose worst tetrahedra have a level of quality substantially better than those produced by any previous method for tetrahedral mesh generation or “mesh cleanup. ” Our goal is to aggressively optimize the worst t ..."
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Cited by 39 (4 self)
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Summary. We present a tetrahedral mesh improvement schedule that usually creates meshes whose worst tetrahedra have a level of quality substantially better than those produced by any previous method for tetrahedral mesh generation or “mesh cleanup. ” Our goal is to aggressively optimize the worst tetrahedra, with speed a secondary consideration. Mesh optimization methods often get stuck in bad local optima (poorquality meshes) because their repertoire of mesh transformations is weak. We employ a broader palette of operations than any previous mesh improvement software. Alongside the best traditional topological and smoothing operations, we introduce a topological transformation that inserts a new vertex (sometimes deleting others at the same time). We describe a schedule for applying and composing these operations that rarely gets stuck in a bad optimum. We demonstrate that all three techniques—smoothing, vertex insertion, and traditional transformations—are substantially more effective than any two alone. Our implementation usually improves meshes so that all dihedral angles are between 31 ◦ and 149 ◦ , or (with a different objective function) between 23 ◦ and 136 ◦. 1
Metric tensors for anisotropic mesh generation
, 2005
"... It has been amply demonstrated that significant improvements in accuracy and efficiency can be gained when a properly chosen anisotropic mesh is used in the numerical solution for a large class of problems which exhibit anisotropic solution features. In practice, an anisotropic mesh is commonly gene ..."
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Cited by 38 (8 self)
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It has been amply demonstrated that significant improvements in accuracy and efficiency can be gained when a properly chosen anisotropic mesh is used in the numerical solution for a large class of problems which exhibit anisotropic solution features. In practice, an anisotropic mesh is commonly generated as a quasiuniform mesh in the metric determined by a tensor specifying the shape, size, orientation of elements. Thus, it is crucial to choose an appropriate metric tensor for anisotropic mesh generation and adaptation. In this paper, we develop a general formula for the metric tensor for use in any spatial dimension. The formulation is based on error estimates for polynomial preserving interpolation on simiplicial elements. Numerical results in twodimensions are presented to demonstrate the ability of the metric tensor to produce anisotropic meshes with correct mesh concentration and good overall quality. The procedure developed in this paper for defining the metric tensor can also be applied to other types of error estimates.
OPTIMAL ANISOTROPIC MESHES FOR MINIMIZING INTERPOLATION ERRORS In L^pnorm
, 2006
"... In this paper, we present a new optimal interpolation error estimate in L p norm (1 ≤ p ≤∞) for finite element simplicial meshes in any spatial dimension. A sufficient condition for a mesh to be nearly optimal is that it is quasiuniform under a new metric defined by a modified Hessian matrix of th ..."
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Cited by 37 (0 self)
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In this paper, we present a new optimal interpolation error estimate in L p norm (1 ≤ p ≤∞) for finite element simplicial meshes in any spatial dimension. A sufficient condition for a mesh to be nearly optimal is that it is quasiuniform under a new metric defined by a modified Hessian matrix of the function to be interpolated. We also give new functionals for the global moving mesh method and obtain optimal monitor functions from the viewpoint of minimizing interpolation error in the L p norm. Some numerical examples are also given to support the theoretical estimates.
A unified convergence analysis for local projection stabilisations applied to the Oseen problem
, 2006
"... The discretisation of the Oseen problem by finite element methods suffers in general from two reasons. First, the discrete infsup (Babuška–Brezzi) condition can be violated. Second, spurious oscillations occur due to the dominating convection. One way to overcome both difficulties is the use of l ..."
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Cited by 34 (7 self)
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The discretisation of the Oseen problem by finite element methods suffers in general from two reasons. First, the discrete infsup (Babuška–Brezzi) condition can be violated. Second, spurious oscillations occur due to the dominating convection. One way to overcome both difficulties is the use of local projection techniques. Studying the local projection method in an abstract setting, we show that the fulfilment of a local infsup condition between approximation and projection spaces allows to construct an interpolation with additional orthogonality properties. Based on this special interpolation, optimal apriori error estimates are shown with error constants independent of the Reynolds number. Applying the general theory, we extend the results of Braack and Burman for the standard twolevel version of the local projection stabilisation to discretisations of arbitrary order on simplices, quadrilaterals, and hexahedra. Moreover, our general theory allows to derive a novel class of local projection stabilisation by enrichment of the approximation spaces. This class of stabilised schemes uses approximation and projection spaces defined on the same mesh and leads to much more compact stencils than in the twolevel approach. Finally, on simplices, the spectral equivalence of the stabilising terms of the local projection method and the subgrid modeling introduced by Guermond is shown. This clarifies the relation of the local projection stabilisation to the variational multiscale approach.
Maximum norm a posteriori error estimation for parabolic problems using elliptic reconstructions
 SIAM J. Numer. Anal
"... Abstract. Residualtype a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reactiondiffusion equations posed in polygonal domains. Linear finite elements are considered on anisotropic triangulations. The error constants are independent of the diameters a ..."
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Cited by 33 (11 self)
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Abstract. Residualtype a posteriori error estimates in the maximum norm are given for singularly perturbed semilinear reactiondiffusion equations posed in polygonal domains. Linear finite elements are considered on anisotropic triangulations. The error constants are independent of the diameters and the aspect ratios of mesh elements and of the small perturbation parameter.
Measuring mesh qualities and application to variational mesh
 SIAM J. Sci. Comput
"... Abstract. The mesh assessment problem is investigated in this paper by taking into account the shape and size of elements and the solution behavior. Three elementwise mesh quality measures characterizing the shape, alignment, and adaptation features of elements are introduced according to the estima ..."
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Cited by 24 (7 self)
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Abstract. The mesh assessment problem is investigated in this paper by taking into account the shape and size of elements and the solution behavior. Three elementwise mesh quality measures characterizing the shape, alignment, and adaptation features of elements are introduced according to the estimates of interpolation error developed on a general mesh. An adaptive mesh is assessed by an overall quality measure defined as a weighted Lebesgue norm of a product of the three elementwise quality measures. It is shown that the overall quality of a mesh is good if the overall mesh quality measure is small or significantly smaller than the socalled roughness measure of the solution, defined as the ratio of two Lebesgue norms of a derivative of the solution. The definition of the overall mesh quality measure comes in such a way that the measure appears in the underlying error bound as the only factor depending substantially on the mesh. As an immediate result, the task of mesh adaptation becomes to control the overall mesh quality. This idea is applied to variational mesh adaptation to develop two functionals, one new and the other related to an existing functional recently developed using the regularity and equidistribution arguments. Numerical experiments are given to demonstrate the ability of the functionals to generate adaptive meshes of good quality.