A posteriori error estimation in finite element analysis

Ainsworth, Mark; Oden, J. T.

doi:10.1016/s0045-7825(96)01107-3

Cited by 1,538 publications

(1,910 citation statements)

References 43 publications

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“…We shall discuss this strategy in detail in Section 4. But for all examples shown below, the Gaussian functions are calculated by F (x) as defined in Equation (1).…”

Section: Initial Surface Meshingmentioning

confidence: 99%

“…In addition to providing a geometric representation of an object for graphical purposes, mesh generation is also in great demand in numerical simulation using finite/boundary element methods and has been extensively studied in both applied mathematics [7] and computational engineering [1,27]. Although different types of meshes may be generated depending on the numerical solvers being employed, we restrict ourselves in this paper to triangular (surface) and tetrahedral (volumetric) meshes.…”

Section: Introductionmentioning

confidence: 99%

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Feature-preserving adaptive mesh generation for molecular shape modeling and simulation

Holst

McCammon

2008

Journal of Molecular Graphics and Modelling

104

122

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We describe a chain of algorithms for molecular surface and volumetric mesh generation. We take as inputs the centers and radii of all atoms of a molecule and the toolchain outputs both triangular and tetrahedral meshes that can be used for molecular shape modeling and simulation. Experiments on a number of molecules are demonstrated, showing that our methods possess several desirable properties: feature-preservation, local adaptivity, high quality, and smoothness (for surface meshes). We also demonstrate an example of molecular simulation using the finite element method and the meshes generated by our method. The approaches presented and their implementations are also applicable to other types of inputs such as 3D scalar volumes and triangular surface meshes with low quality, and hence can be used for generation/improvment of meshes in a broad range of applications.

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“…We shall discuss this strategy in detail in Section 4. But for all examples shown below, the Gaussian functions are calculated by F (x) as defined in Equation (1).…”

Section: Initial Surface Meshingmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Feature-preserving adaptive mesh generation for molecular shape modeling and simulation

Holst

McCammon

2008

Journal of Molecular Graphics and Modelling

104

122

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“…In this case, the system (2.13) does not have a unique solution. However, u GF EM is unique, i.e., if {c (1) ji } and {c (2) ji } are two solutions of (2.13), then…”

Section: Generalized Finite Element Methodsmentioning

confidence: 99%

“…An extensive bibliography (before 1998) on superconvergence is available in [20], where as many references on 3-dimensional problems can be found in [19]. Moreover, there have been several books written on superconvergence in the context of the finite element method, e.g., [1,6,13,14,23,35,38].…”

Section: Introductionmentioning

confidence: 99%

Superconvergence in the Generalized Finite Element Method

Babuška¹,

Banerjee²,

Osborn³

2005

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In this paper, we address the problem of the existence of superconvergence points of approximate solutions, obtained from the Generalized Finite Element Method (GFEM), of a Neumann elliptic boundary value problem. GFEM is a Galerkin method that uses non-polynomial shape functions, and was developed in [4,5,24]. In particular, we show that the superconvergence points for the gradient of the approximate are zeros of certain systems of non-linear equations that do not depend on the solution of the boundary value problem. For approximate solutions with second derivatives, we have also characterized the superconvergence points of the second derivatives of the approximate solution as the roots of certain systems of non-linear equations. We note that it is easy to construct smooth generalized finite element approximation.

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“…In such analysis one tries to determine computable quantities that can be used to guide an adaptive procedure. For an overview of a posteriori error analyses and their applications, we refer to the seminal papers of Babuška and Rheinboldt [BR78a,BR78b], the books of Ainsworth and Oden [AO00] and Verfürth [Ver96], and the more recent developments of Dörfler [Dör96] and Morin et al [MNS00,MNS].…”

Section: Introductionmentioning

confidence: 99%

On the a posteriori error analysis for equations of prescribed mean curvature

Fierro

Veeser

2003

Math. Comp.

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Abstract. We present two approaches to the a posteriori error analysis for prescribed mean curvature equations. The main difference between them concerns the estimation of the residual: without or with computable weights. In the second case, the weights are related to the eigenvalues of the underlying operator and thus provide local and computable information about the conditioning. We analyze the two approaches from a theoretical viewpoint. Moreover, we investigate and compare the performance of the derived indicators in an adaptive procedure. Our theoretical and practical results show that it is advantageous to estimate the residual in a weighted way.

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A posteriori error estimation in finite element analysis

Cited by 1,538 publications

References 43 publications

Feature-preserving adaptive mesh generation for molecular shape modeling and simulation

Feature-preserving adaptive mesh generation for molecular shape modeling and simulation

Superconvergence in the Generalized Finite Element Method

On the a posteriori error analysis for equations of prescribed mean curvature

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