Generalized Finite Element Methods — Main Ideas, Results and Perspective

Babuška, Ivo; Banerjee, Uday; Osborn, John E.

doi:10.1142/s0219876204000083

Cited by 220 publications

(180 citation statements)

References 36 publications

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“…Babuška et al [2] use the phrase special finite elements to denote finite element methods (FEM) that employ basis functions that, for instance, incorporate specialized knowledge of the partial differential operator. Many methods have been proposed to incorporate relevant information into the special basis functions; for instance the generalized FEM (GFEM) [4] and the multiscale FEM (MsFEM) [9]. The purpose of this section is to compare the special finite element introduced in Section 3 for the solution of (1.1) with the classical FEM, MsFEM, and GFEM.…”

Section: Relationship To Other Approximating Methodsmentioning

confidence: 99%

“…φ j ξ j will belong to H 1 0 (Ω). If, in addition, the functions φ j and their gradients ∇φ j are uniformly bounded, Babuška et al [4] prove convergence estimates 6 for GFEM. In order to show that a special finite element method is a GFEM, we need to exhibit patches {ω j }, the partition of unity {φ j }, and subspaces S 1 , .…”

Section: Gfemmentioning

confidence: 99%

“…Motivated by this orthogonal decomposition, we develop a conforming finite dimensional approximation space. We contrast our CMS-based approach with the multiscale finite element method (MsFEM) [9] and draw a relationship with the generalized finite element method (GFEM) [4]. We argue that our approach is not a generalized finite element method, in contrast to MsFEM.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A special finite element method based on component mode synthesis

Hetmaniuk¹,

Lehoucq²

2010

ESAIM: M2AN

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Abstract. The goal of our paper is to introduce basis functions for the finite element discretization of a second order linear elliptic operator with rough or highly oscillating coefficients. The proposed basis functions are inspired by the classic idea of component mode synthesis and exploit an orthogonal decomposition of the trial subspace to minimize the energy. Numerical experiments illustrate the effectiveness of the proposed basis functions.Mathematics Subject Classification. 35J20, 65F15, 65N25, 65N30, 65N55.

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Section: Relationship To Other Approximating Methodsmentioning

confidence: 99%

Section: Gfemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A special finite element method based on component mode synthesis

Hetmaniuk¹,

Lehoucq²

2010

ESAIM: M2AN

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“…Neglecting the time derivatives in equation (1) and assuming di usivity and thermal conductivity as constants D and K , respectively, allows to solve the equations analytically. With analytic expressions for water vapor density ρv and temperature T, one can calculate the normal derivatives arising in the surface integrals in (2). Using the assumption of a spherical hydrometeor with radius rω again, the integrals can be evaluated exactly and one arrives at …”

Section: Maxwellian Growth Equationsmentioning

confidence: 99%

Local Interactions by Diffusion between Mixed-Phase Hydrometeors: Insights from Model Simulations

Baumgartner¹,

Spichtinger²

2017

Mathematics of Climate and Weather Forecasting

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Di usion of water vapor is the dominant growth mechanism for small water droplets and ice crystals in clouds. In current cloud models, Maxwell's theory is used for describing growth of cloud particles. In this approach the local interaction between particles is neglected; the particles can only grow due to changes in environmental conditions, which are assumed as boundary conditions at in nity. This assumption is meaningful if the particles are well separated and far away from each other. However, turbulent motions might change the distances between cloud particles and thus these particles are no longer well separated leading to direct local interactions. In this study we develop a reference model for investigating the direct interaction of cloud particles in mixed-phase clouds as driven by di usion processes. The model is numerically integrated using nite elements. Additionally, we develop a numerical method based on generalized nite elements for including moving particles and their direct interactions with respect to di usional growth and evaporation. Several idealized simulations are carried out for investigating direct interactions of liquid droplets and ice particles in a mixed-phase cloud. The results show that local interaction between cloud particles might enhance life times of droplets and ice particles and thus lead to changes in mixed-phase cloud life time and properties.

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“…Irregular datasets -or unstructured grids -, are mainly used for simulations, for example for finite element analysis [2]. Rendering methods for such grids are an ongoing field of research.…”

Section: Introductionmentioning

confidence: 99%

Volumetric Real-Time Particle-Based Representation of Large Unstructured Tetrahedral Polygon Meshes

Voglreiter

Steinberger

Schmalstieg

et al. 2012

Lecture Notes in Computer Science

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Abstract. In this paper we propose a particle-based volume rendering approach for unstructured, three-dimensional, tetrahedral polygon meshes. We stochastically generate millions of particles per second and project them on the screen in real-time. In contrast to previous rendering techniques of tetrahedral volume meshes, our method does not need a prior depth sorting of geometry. Instead, the rendered image is generated by choosing particles closest to the camera. Furthermore, we use spatial superimposing. Each pixel is constructed from multiple subpixels. This approach not only increases projection accuracy, but allows also a combination of subpixels into one superpixel that creates the well-known translucency effect of volume rendering. We show that our method is fast enough for the visualization of unstructured three-dimensional grids with hard real-time constraints and that it scales well for a high number of particles.

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Generalized Finite Element Methods — Main Ideas, Results and Perspective

Cited by 220 publications

References 36 publications

A special finite element method based on component mode synthesis

A special finite element method based on component mode synthesis

Local Interactions by Diffusion between Mixed-Phase Hydrometeors: Insights from Model Simulations

Volumetric Real-Time Particle-Based Representation of Large Unstructured Tetrahedral Polygon Meshes

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