900 research outputs found
Composable code generation for high order, compatible finite element methods
It has been widely recognised in the HPC communities across the world, that exploiting modern
computer architectures, including exascale machines, to a full extent requires software commu-
nities to adapt their algorithms. Computational methods with a high ratio of floating point op-
erations to bandwidth are favorable. For solving partial differential equations, which can model
many physical problems, high order finite element methods can calculate approximations with a
high efficiency when a good solver is employed. Matrix-free algorithms solve the corresponding
equations with a high arithmetic intensity. Vectorisation speeds up the operations by calculating
one instruction on multiple data elements.
Another recent development for solving partial differential are compatible (mimetic) finite ele-
ment methods. In particular with application to geophysical flows, compatible discretisations ex-
hibit desired numerical properties required for accurate approximations. Among others, this has
been recognised by the UK Met office and their new dynamical core for weather and climate fore-
casting is built on a compatible discretisation. Hybridisation has been proven to be an efficient
solver for the corresponding equation systems, because it removes some inter-elemental coupling
and localises expensive operations.
This thesis combines the recent advances on vectorised, matrix-free, high order finite element
methods in the HPC community on the one hand and hybridised, compatible discretisations in
the geophysical community on the other. In previous work, a code generation framework has been
developed to support the localised linear algebra required for hybridisation. First, the framework
is adapted to support vectorisation and further, extended so that the equations can be solved fully
matrix-free. Promising performance results are completing the thesis.Open Acces
Adaptive Discontinuous Galerkin Finite Element Methods
The Discontinuous Galerkin Method is one variant of the Finite Element Methods for solving partial differential equations, which was first introduced by Reed and Hill in 1970’s [27]. Discontinuous GalerkinMethod (DGFEM) differs from the standard Galerkin FEMthat continuity constraints are not imposed on the inter-element boundaries. It results in a solution which is composed of totally piecewise discontinuous functions. The absence of continuity constraints on the inter-element boundaries implies that DG method has a great deal of flexibility at the cost of increasing the number of degrees of freedom. This flexibility is the source of many but not all of the advantages of the DGFEM method over the Continuous Galerkin (CGFEM) method that uses spaces of continuous piecewise polynomial functions and other ”less standard” methods such as nonconforming methods. As DGFEM method leads to bigger system to solve, theoretical and practical approaches to speed it up are our main focus in this dissertation. This research aims at designing and building an adaptive discontinuous Galerkin finite element method to solve partial differential equations with fast time for desired accuracy on modern architecture
Efficient resonance computations for Helmholtz problems based on a Dirichlet-to-Neumann map
We present an efficient procedure for computing resonances and resonant modes
of Helmholtz problems posed in exterior domains. The problem is formulated as a
nonlinear eigenvalue problem (NEP), where the nonlinearity arises from the use
of a Dirichlet-to-Neumann map, which accounts for modeling unbounded domains.
We consider a variational formulation and show that the spectrum consists of
isolated eigenvalues of finite multiplicity that only can accumulate at
infinity. The proposed method is based on a high order finite element
discretization combined with a specialization of the Tensor Infinite Arnoldi
method. Using Toeplitz matrices, we show how to specialize this method to our
specific structure. In particular we introduce a pole cancellation technique in
order to increase the radius of convergence for computation of eigenvalues that
lie close to the poles of the matrix-valued function. The solution scheme can
be applied to multiple resonators with a varying refractive index that is not
necessarily piecewise constant. We present two test cases to show stability,
performance and numerical accuracy of the method. In particular the use of a
high order finite element discretization together with TIAR results in an
efficient and reliable method to compute resonances
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Schnelle Löser für Partielle Differentialgleichungen
The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch (Leipzig), and Gabriel Wittum (Frankfurt am Main), was held May 22nd–May 28th, 2011. This meeting was well attended by 54 participants with broad geographic representation from 7 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds
A parallel multigrid solver for multi-patch Isogeometric Analysis
Isogeometric Analysis (IgA) is a framework for setting up spline-based
discretizations of partial differential equations, which has been introduced
around a decade ago and has gained much attention since then. If large spline
degrees are considered, one obtains the approximation power of a high-order
method, but the number of degrees of freedom behaves like for a low-order
method. One important ingredient to use a discretization with large spline
degree, is a robust and preferably parallelizable solver. While numerical
evidence shows that multigrid solvers with standard smoothers (like Gauss
Seidel) does not perform well if the spline degree is increased, the multigrid
solvers proposed by the authors and their co-workers proved to behave optimal
both in the grid size and the spline degree. In the present paper, the authors
want to show that those solvers are parallelizable and that they scale well in
a parallel environment.Comment: The first author would like to thank the Austrian Science Fund (FWF)
for the financial support through the DK W1214-04, while the second author
was supported by the FWF grant NFN S117-0
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