Mathematisches Seminar

des DFG-Sonderforschungsbereichs 393

Numerische Simulation auf massiv parallelen Rechnern

Zeit: Freitag, 07.04.2000, 11:45 Uhr

Ort: Reichenhainer Straße 70, B 202

Vortragender: I. Graham (Bath)

Thema: Divergence-free mixed finite elements for groundwater flow problems

In this talk we describe recent work on iterative methods for solving mixed finite element discretisations of second order elliptic problems, with emphasis on applications in groundwater flow problems, with highly variable permeability.
Discretisation using the lowest order Raviart-Thomas elements leads to ill-conditioned saddle point systems for velocity/pressure. We describe a method of computing the velocity by solving an SPD system, decoupled from the pressure. This requires the construction of a basis for the subspace of divergence-free Raviart-Thomas elements. In 2D (resp. 3D) the resulting velocity system is a fifth (resp. a third) of the size of the original indefinite system. In particular, in 2D for any boundary conditions, the reduced SPD problem turns out to be a bordered system with main block consisting of the discretisation of a 2nd-order elliptic problem for the stream function with standard linear H¹ elements, for which the theory of preconditioning is well understood.
We describe in detail the use of this technique as a solver in the parallel simulation of flow in 2D heterogeneous media. Here the chief numerical difficulty is that the permeability takes a different value on each element, and may vary by many orders of magnitude from element to element, especially when the underlying statistical model has a large variance and a small correlation length. Lack of regularity in the problem means that systems of size 10⁶ - 10⁸ must be solved in practice. We solve the reduced SPD systems for this problem using a parallel domain decomposition-CG code.
We also indicate recent results of R. Scheichl on extension of the reduction technique to three dimensional problems.
This talk represents joint work with K. A. Cliffe, R. Scheichl and L. Stals.

Das Seminar wird von Herrn Dr. Apel geleitet. Interessenten sind herzlich eingeladen.

Thomas Apel,

Zeit:	Freitag, 07.04.2000, 11:45 Uhr
Ort:	Reichenhainer Straße 70, B 202
Vortragender:	I. Graham (Bath)
Thema:	Divergence-free mixed finite elements for groundwater flow problems
In this talk we describe recent work on iterative methods for solving mixed finite element discretisations of second order elliptic problems, with emphasis on applications in groundwater flow problems, with highly variable permeability. Discretisation using the lowest order Raviart-Thomas elements leads to ill-conditioned saddle point systems for velocity/pressure. We describe a method of computing the velocity by solving an SPD system, decoupled from the pressure. This requires the construction of a basis for the subspace of divergence-free Raviart-Thomas elements. In 2D (resp. 3D) the resulting velocity system is a fifth (resp. a third) of the size of the original indefinite system. In particular, in 2D for any boundary conditions, the reduced SPD problem turns out to be a bordered system with main block consisting of the discretisation of a 2nd-order elliptic problem for the stream function with standard linear H¹ elements, for which the theory of preconditioning is well understood. We describe in detail the use of this technique as a solver in the parallel simulation of flow in 2D heterogeneous media. Here the chief numerical difficulty is that the permeability takes a different value on each element, and may vary by many orders of magnitude from element to element, especially when the underlying statistical model has a large variance and a small correlation length. Lack of regularity in the problem means that systems of size 10⁶ - 10⁸ must be solved in practice. We solve the reduced SPD systems for this problem using a parallel domain decomposition-CG code. We also indicate recent results of R. Scheichl on extension of the reduction technique to three dimensional problems. This talk represents joint work with K. A. Cliffe, R. Scheichl and L. Stals.
Das Seminar wird von Herrn Dr. Apel geleitet. Interessenten sind herzlich eingeladen.

Mathematisches Seminar

des DFG-Sonderforschungsbereichs 393 Numerische Simulation auf massiv parallelen Rechnern

des DFG-Sonderforschungsbereichs 393

Numerische Simulation auf massiv parallelen Rechnern