A comparison of adaptive refinement techniques for elliptic problems

Mitchell, William F.

doi:10.1145/76909.76912

Cited by 169 publications

(103 citation statements)

References 11 publications

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“…2 for a simple case. The bisection of paired triangles was first introduced by Mitchell [60,61]. The idea was generalized by Kossaczký [56] to three dimensions, and Maubach [59] and Stevenson [80] to any dimension.…”

Section: Compatible Bisectionsmentioning

confidence: 99%

Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids

Chen

Nochetto

2009

Multiscale, Nonlinear and Adaptive Approximation

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We give an overview of multilevel methods, such as V-cycle multigrid and BPX preconditioner, for solving various partial differential equations (including H(grad), H(curl) and H(div) systems) on quasi-uniform meshes and extend them to graded meshes and completely unstructured grids. We first discuss the classical multigrid theory on the basis of the method of subspace correction of Xu and a key identity of Xu and Zikatanov. We next extend the classical multilevel methods in H(grad) to graded bisection grids upon employing the decomposition of bisection grids of Chen, Nochetto, and Xu. We finally discuss a class of multilevel preconditioners developed by Hiptmair and Xu for problems discretized on unstructured grids and extend them to H(curl) and H(div) systems over graded bisection grids.

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Section: Compatible Bisectionsmentioning

confidence: 99%

Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids

Chen

Nochetto

2009

Multiscale, Nonlinear and Adaptive Approximation

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“…The bisection of paired triangles was first introduced by Mitchell for dimension d = 2 [37,38]. The idea was generalized by Kossaczký [32] to d = 3, and Maubach [33] and Stevenson [54] to d ≥ 2.…”

Section: Compatible Bisectionsmentioning

confidence: 99%

“…McCormick and collaborators [34,35] developed the fast adaptive composite grid (FAC) method, which requires exact solvers on subdomains that are partitioned by uniform grids; hence mesh adaptivity is achieved via superposition of tensor-product rectangular grids. Rivara [47] and Mitchell [37,38] developed local multigrid methods on adaptive triangular grids obtained by longest edge bisection and newest vertex bisection, respectively, for d = 2. Also for d = 2, Bank, Sherman and Weiser [7] proposed the red-green refinement strategy, which was implemented in the wellknown piecewise linear triangular multigrid software package (PLTMG) of Bank [5].…”

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confidence: 99%

Optimal multilevel methods for graded bisection grids

Chen

Nochetto

2011

Numer. Math.

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Abstract. We design and analyze optimal additive and multiplicative multilevel methods for solving H 1 problems on graded grids obtained by bisection. We deal with economical local smoothers: after a global smoothing in the finest mesh, local smoothing for each added node during the refinement needs to be performed only for three vertices -the new vertex and its two parent vertices. We show that our methods lead to optimal complexity for any dimensions and polynomial degree. The theory hinges on a new decomposition of bisection grids in any dimension, which is of independent interest and yields a corresponding decomposition of spaces. We use the latter to bridge the gap between graded and quasi-uniform grids, for which the multilevel theory is well-established.

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“…While the general CFE method can effectively be combined with an adaptive mesh refinement (see e.g. [54,23]), in our case the domain description via image data naturally defines the finest computational mesh as the one associated with the 3D image data. Far from the interface, the CFE basis functions of our approach coincide with the standard basis functions on an overlaid structured grid.…”

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confidence: 99%

3D Composite Finite Elements for Elliptic Boundary Value Problems with Discontinuous Coefficients

Preußer¹,

Rumpf

Sauter³

et al. 2011

SIAM J. Sci. Comput.

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Abstract. For scalar and vector-valued elliptic boundary value problems with discontinuous coefficients across geometrically complicated interfaces, a composite finite element approach is developed. Composite basis functions are constructed, mimicking the expected jump condition for the solution at the interface in an approximate sense. The construction is based on a suitable local interpolation on the space of admissible functions. We study the order of approximation and the convergence properties of the method numerically. As applications, heat diffusion in an aluminum foam matrix filled with polymer and linear elasticity of micro-structured materials, in particular specimens of trabecular bone, are investigated. Furthermore, a numerical homogenization approach is developed for periodic structures and real material specimens which are not strictly periodic but are considered as statistical prototypes. Thereby, effective macroscopic material properties can be computed.Key words. composite finite elements, homogenization, elliptic partial differential equations, discontinuous coefficients AMS subject classifications. 65M60, 65N30, 74S05, 74Q05, 80M401. Introduction. Simulations in materials science or bio-medical applications are frequently faced with multi-phase materials having interfaces of complicated structure. Examples are heat conduction in chip design [26], the elastic behavior of composite materials [59], electric fields in the human body [85] in the context of electrocardiography [30], the brain shift in neurosurgery [82], and effects of vertebroplasty on macroscopic properties of trabecular microstructure [46]. The standard finite element (FE) procedure in this context is to generate a geometrically complicated simplicial (i.e. triangular or tetrahedral in 2D or 3D, respectively) FE mesh that resolves the interface between the different materials. On these meshes standard FE basis functions are used for the discretization of the physical quantities. However, generating 3D meshes suitable for FE simulations is difficult [13,75,69] and may require substantial user interaction.Composite Finite Elements. Composite finite elements (CFE) are based on the idea of incorporating the geometric complexity of physical domains [34,33,35,61] or interfaces between subdomains with different material properties [65] into the shape of basis functions rather than in the FE mesh. A corresponding multigrid method has been investigated in [65]. The term 'composite' has also appeared in the FE literature in Composite Triangles [31,76]. Like our approach presented here, these methods also use a virtual subdivision of tetrahedral elements, however, not as an adaptation to the geometry of the underlying domains.The approach presented in this paper is based on [65] for 2D problems. We extend this construction to 3D anisotropic PDE and as a vector-valued problem to 3D linearized elasticity. In fact, we construct the composite element method in case of level set described domains, where the level set functions is given an underly...

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A comparison of adaptive refinement techniques for elliptic problems

Cited by 169 publications

References 11 publications

Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids

Optimal multilevel methods for H(grad), H(curl), and H(div) systems on graded and unstructured grids

Optimal multilevel methods for graded bisection grids

3D Composite Finite Elements for Elliptic Boundary Value Problems with Discontinuous Coefficients

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