Composite finite elements for the approximation of PDEs on domains with complicated micro-structures

Hackbusch, Wolfgang; Sauter, Stéfan A.

doi:10.1007/s002110050248

Cited by 121 publications

(127 citation statements)

References 4 publications

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“…In the following section we use the virtual grid G for the construction of a CFE space for problems on domains with complicated boundaries (cf. [20]). The multigrid method presented in Section 3 leads to convergence rates which are independent of the grid width h. Later, in Section 5 we present an outlook for the use of the virtual grid to define a CFE space for problems with discontinuous coefficients.…”

Section: A Virtual Gridmentioning

confidence: 99%

“…Below we describe how this scaling is performed in the case of mass and stiffness matrices. The entries of the local matrix are weighted and accumulated into the global one (lines [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23].…”

Section: Matrix Assemblymentioning

confidence: 99%

“…Our approach is inspired by [20] for composite finite elements (CFE) in 2D. It combines the idea of CFE with the efficiency of structured grids as they are used in image processing [36].…”

Section: Introductionmentioning

confidence: 99%

“…Instead, they build the necessary adaption into basis functions, which are then used in a standard Galerkin method [20]. Far from the complicated structures (domain boundaries or interfaces between internal object structures), the basis functions coincide with the standard basis functions on the structured grid.…”

Section: Introductionmentioning

confidence: 99%

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Composite finite elements for 3D image based computing

Liehr

Preußer

Rumpf

et al. 2008

Comput. Visual Sci.

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Abstract. We present an algorithmical concept for modeling and simulation with partial differential equations (PDEs) in image based computing where the computational geometry is defined through previously segmented image data. Such problems occur in applications from biology and medicine where the underlying image data has been acquired through e. g. computed tomography (CT), magnetic resonance imaging (MRI) or electron microscopy (EM). Based on a level-set description of the computational domain, our approach is capable of automatically providing suitable composite finite element functions that resolve the complicated shapes in the medical/biological data set. It is efficient in the sense that the traversal of the grid (and thus assembling matrices for finite element computations) inherits the efficiency of uniform grids away from complicated structures. The method's efficiency heavily depends on precomputed lookup tables in the vicinity of the domain boundary or interface. A suitable multigrid method is used for an efficient solution of the systems of equations resulting from the composite finite element discretization. The paper focuses on both algorithmical and implementational details. Scalar and vector valued model problems as well as real applications underline the usability of our approach.

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Section: A Virtual Gridmentioning

confidence: 99%

Section: Matrix Assemblymentioning

confidence: 99%

“…Our approach is inspired by [20] for composite finite elements (CFE) in 2D. It combines the idea of CFE with the efficiency of structured grids as they are used in image processing [36].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Composite finite elements for 3D image based computing

Liehr

Preußer

Rumpf

et al. 2008

Comput. Visual Sci.

View full text Add to dashboard Cite

show abstract

“…Thus, from the viewpoint of accuracy and balancing of local errors, we cannot expect that the degrees of freedom of such a finite element space are distributed in a (nearly) optimal way. In [HS1], composite finite element spaces have been introduced where the minimal number of unknowns is independent of the size and number of geometric details. The combination of composite finite element spaces with an a posteriori error estimator (used as an error indicator) allows to design problem-adapted finite element spaces where the adaptation process starts from very coarse levels.…”

Section: Introductionmentioning

confidence: 99%

A posteriori error analysis for elliptic pdes on domains with complicated structures

Carstensen

Sauter

2004

Numerische Mathematik

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The discretisation of boundary value problems on complicated domains cannot resolve all geometric details such as small holes or pores. The model problem of this paper consists of a triangulated polygonal domain with holes of a size of the mesh-width at most and mixed boundary conditions for the Poisson equation. Reliable and efficient a posteriori error estimates are presented for a fully numerical discretisation with conforming piecewise affine finite elements. Emphasis is on technical difficulties with the numerical approximation of the domain and their influence on the constants in the reliability and efficiency estimates.

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Algebraic multigrid methods for the solution of the Navier-Stokes equations in complicated geometries

Griebel

Neunhoeffer

Regler

1998

Int. J. Numer. Meth. Fluids

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The application of standard multigrid methods for the solution of the Navier-Stokes equations in complicated domains causes problems in two ways. First, coarsening is not possible to full extent since the geometry must be resolved by the coarsest grid used, and second, for semiimplicit time stepping schemes, robustness of the convergence rates is usually not obtained for the arising convection-diffusion problems, especially for higher Reynolds numbers.We show that both problems can be overcome by the use of algebraic multigrid (AMG) which we apply for the solution of the pressure and momentum equations in explicit and semiimplicit time-stepping schemes.We consider the convergence rates of AMG for several model problems and we demonstrate the robustness of the proposed scheme.

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Composite finite elements for the approximation of PDEs on domains with complicated micro-structures

Cited by 121 publications

References 4 publications

Composite finite elements for 3D image based computing

Composite finite elements for 3D image based computing

A posteriori error analysis for elliptic pdes on domains with complicated structures

Algebraic multigrid methods for the solution of the Navier-Stokes equations in complicated geometries

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