3 research outputs found
A preconditioning strategy for linear systems arising from nonsymmetric schemes in isogeometric analysis
In the context of isogeometric analysis, we consider two discretization
approaches that make the resulting stiffness matrix nonsymmetric even if the
differential operator is self-adjoint. These are the collocation method and the
recently-developed weighted quadrature for the Galerkin discretization. In this
paper, we are interested in the solution of the linear systems arising from the
discretization of the Poisson problem using one of these approaches. In [SIAM
J. Sci. Comput. 38(6) (2016) pp. A3644--A3671], a well-established direct
solver for linear systems with tensor structure was used as a preconditioner in
the context of Galerkin isogeometric analysis, yielding promising results. In
particular, this preconditioner is robust with respect to the mesh size and
the spline degree . In the present work, we discuss how a similar approach
can applied to the considered nonsymmetric linear systems. The efficiency of
the proposed preconditioning strategy is assessed with numerical experiments on
two-dimensional and three-dimensional problems
Robust Preconditioning for Space-Time Isogeometric Analysis of Parabolic Evolution Problems
We propose and investigate new robust preconditioners for space-time
Isogeometric Analysis of parabolic evolution problems. These preconditioners
are based on a time parallel multigrid method. We consider a decomposition of
the space-time cylinder into time-slabs which are coupled via a discontinuous
Galerkin technique. The time-slabs provide the structure for the time-parallel
multigrid solver. The most important part of the multigrid method is the
smoother. We utilize the special structure of the involved operator to decouple
its application into several spatial problems by means of generalized
eigenvalue or Schur decompositions. Some of these problems have a symmetric
saddle point structure, for which we present robust preconditions. Finally, we
present numerical experiments confirming the robustness of our space-time IgA
solver
Efficient Direct Space-Time Finite Element Solvers for Parabolic Initial-Boundary Value Problems in Anisotropic Sobolev Spaces
We consider a space-time variational formulation of parabolic
initial-boundary value problems in anisotropic Sobolev spaces in combination
with a Hilbert-type transformation. This variational setting is the starting
point for the space-time Galerkin finite element discretization that leads to a
large global linear system of algebraic equations. We propose and investigate
new efficient direct solvers for this system. In particular, we use a
tensor-product approach with piecewise polynomial, globally continuous ansatz
and test functions. The developed solvers are based on the Bartels-Stewart
method and on the Fast Diagonalization method, which result in solving a
sequence of spatial subproblems. The solver based on the Fast Diagonalization
method allows to solve these spatial subproblems in parallel leading to a full
parallelization in time. We analyze the complexity of the proposed algorithms,
and give numerical examples for a two-dimensional spatial domain, where sparse
direct solvers for the spatial subproblems are used