569 research outputs found
Schnelle Löser für partielle Differentialgleichungen
The workshop Schnelle Löser für partielle Differentialgleichungen, organised by Randolph E. Bank (La Jolla), Wolfgang Hackbusch(Leipzig), Gabriel Wittum (Heidelberg) was held May 22nd - May 28th, 2005. This meeting was well attended by 47 participants with broad geographic representation from 9 countries and 3 continents. This workshop was a nice blend of researchers with various backgrounds
Compressive Space-Time Galerkin Discretizations of Parabolic Partial Differential Equations
We study linear parabolic initial-value problems in a space-time variational
formulation based on fractional calculus. This formulation uses "time
derivatives of order one half" on the bi-infinite time axis. We show that for
linear, parabolic initial-boundary value problems on , the
corresponding bilinear form admits an inf-sup condition with sparse tensor
product trial and test function spaces. We deduce optimality of compressive,
space-time Galerkin discretizations, where stability of Galerkin approximations
is implied by the well-posedness of the parabolic operator equation. The
variational setting adopted here admits more general Riesz bases than previous
work; in particular, no stability in negative order Sobolev spaces on the
spatial or temporal domains is required of the Riesz bases accommodated by the
present formulation. The trial and test spaces are based on Sobolev spaces of
equal order with respect to the temporal variable. Sparse tensor products
of multi-level decompositions of the spatial and temporal spaces in Galerkin
discretizations lead to large, non-symmetric linear systems of equations. We
prove that their condition numbers are uniformly bounded with respect to the
discretization level. In terms of the total number of degrees of freedom, the
convergence orders equal, up to logarithmic terms, those of best -term
approximations of solutions of the corresponding elliptic problems.Comment: 26 page
A parallel algorithm for solving linear parabolic evolution equations
We present an algorithm for the solution of a simultaneous space-time
discretization of linear parabolic evolution equations with a symmetric
differential operator in space. Building on earlier work, we recast this
discretization into a Schur-complement equation whose solution is a
quasi-optimal approximation to the weak solution of the equation at hand.
Choosing a tensor-product discretization, we arrive at a remarkably simple
linear system. Using wavelets in time and standard finite elements in space, we
solve the resulting system in linear complexity on a single processor, and in
polylogarithmic complexity when parallelized in both space and time. We
complement these theoretical findings with large-scale parallel computations
showing the effectiveness of the method
Self-adaptive isogeometric spatial discretisations of the first and second-order forms of the neutron transport equation with dual-weighted residual error measures and diffusion acceleration
As implemented in a new modern-Fortran code, NURBS-based isogeometric analysis (IGA) spatial discretisations and self-adaptive mesh refinement (AMR) algorithms are developed in the application to the first-order and second-order forms of the neutron transport equation (NTE).
These AMR algorithms are shown to be computationally efficient and numerically accurate when compared to standard approaches. IGA methods are very competitive and offer certain unique advantages over standard finite element methods (FEM), not least of all because the numerical analysis is performed over an exact representation of the underlying geometry, which is generally available in some computer-aided design (CAD) software description. Furthermore, mesh refinement can be performed within the analysis program at run-time, without the need to revisit any ancillary mesh generator. Two error measures are described for the IGA-based AMR algorithms, both of which can be employed in conjunction with energy-dependent meshes. The first heuristically minimises any local contributions to the global discretisation error, as per some appropriate user-prescribed norm. The second employs duality arguments to minimise important local contributions to the error as measured in some quantity of interest; this is commonly known as a dual-weighted residual (DWR) error measure and it demands the solution to both the forward (primal) and the adjoint (dual) NTE.
Finally, convergent and stable diffusion acceleration and generalised minimal residual (GMRes) algorithms, compatible with the aforementioned AMR algorithms, are introduced to accelerate the convergence of the within-group self-scattering sources for scattering-dominated problems for the first and second-order forms of the NTE. A variety of verification benchmark problems are analysed to demonstrate the computational performance and efficiency of these acceleration techniques.Open Acces
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