10 research outputs found
Robust isogeometric preconditioners for the Stokes system based on the Fast Diagonalization method
In this paper we propose a new class of preconditioners for the isogeometric
discretization of the Stokes system. Their application involves the solution of
a Sylvester-like equation, which can be done efficiently thanks to the Fast
Diagonalization method. These preconditioners are robust with respect to both
the spline degree and mesh size. By incorporating information on the geometry
parametrization and equation coefficients, we maintain efficiency on
non-trivial computational domains and for variable kinematic viscosity. In our
numerical tests we compare to a standard approach, showing that the overall
iterative solver based on our preconditioners is significantly faster.Comment: 31 pages, 4 figure
Optimal-order isogeometric collocation at Galerkin superconvergent points
In this paper we investigate numerically the order of convergence of an
isogeometric collocation method that builds upon the least-squares collocation
method presented in [1] and the variational collocation method presented in
[2]. The focus is on smoothest B-splines/NURBS approximations, i.e, having
global continuity for polynomial degree . Within the framework of
[2], we select as collocation points a subset of those considered in [1], which
are related to the Galerkin superconvergence theory. With our choice, that
features local symmetry of the collocation stencil, we improve the convergence
behaviour with respect to [2], achieving optimal -convergence for odd
degree B-splines/NURBS approximations. The same optimal order of convergence is
seen in [1], where, however a least-squares formulation is adopted. Further
careful study is needed, since the robustness of the method and its
mathematical foundation are still unclear.Comment: 21 pages, 20 figures (35 pdf images
Space-time least-squares isogeometric method and efficient solver for parabolic problems
In this paper, we propose a space-time least-squares isogeometric method to
solve parabolic evolution problems, well suited for high-degree smooth splines
in the space-time domain. We focus on the linear solver and its computational
efficiency: thanks to the proposed formulation and to the tensor-product
construction of space-time splines, we can design a preconditioner whose
application requires the solution of a Sylvester-like equation, which is
performed efficiently by the fast diagonalization method. The preconditioner is
robust w.r.t. spline degree and mesh size. The computational time required for
its application, for a serial execution, is almost proportional to the number
of degrees-of-freedom and independent of the polynomial degree. The proposed
approach is also well-suited for parallelization.Comment: 29 pages, 8 figure
A low-rank isogeometric solver based on Tucker tensors
We propose an isogeometric solver for Poisson problems that combines i)
low-rank tensor techniques to approximate the unknown solution and the system
matrix, as a sum of a few terms having Kronecker product structure, ii) a
Truncated Preconditioned Conjugate Gradient solver to keep the rank of the
iterates low, and iii) a novel low-rank preconditioner, based on the Fast
Diagonalization method where the eigenvector multiplication is approximated by
the Fast Fourier Transform. Although the proposed strategy is written in
arbitrary dimension, we focus on the three-dimensional case and adopt the
Tucker format for low-rank tensor representation, which is well suited in low
dimension. We show in numerical tests that this choice guarantees significant
memory saving compared to the full tensor representation. We also extend and
test the proposed strategy to linear elasticity problems.Comment: 27 pages, 8 figure
A domain decomposition method for isogeometric multi-patch problems with inexact local solvers
In Isogeometric Analysis, the computational domain is often described as
multi-patch, where each patch is given by a tensor product spline/NURBS
parametrization. In this work we propose a FETI-like solver where local inexact
solvers exploit the tensor product structure at the patch level. To this
purpose, we extend to the isogeometric framework the so-called All-Floating
variant of FETI, that allows us to use the Fast Diagonalization method at the
patch level. We construct then a preconditioner for the whole system and prove
its robustness with respect to the local mesh-size and patch-size
(i.e., we have scalability). Our numerical tests confirm the theory and also
show a favourable dependence of the computational cost of the method from the
spline degree .Comment: 19 pages, 1 figur
The virtual element method on image-based domain approximations
We analyze and validate the virtual element method combined with a projection
approach similar to the one in [1, 2], to solve problems on two dimensional
domains with curved boundaries approximated by polygonal domains obtained as
the union of squared elements out of a uniform structured mesh, such as the one
that naturally arise when the domain is issued from an image. We show, both
theoretically and numerically, that resorting to the use of polygonal element
allows to satisfy the assumptions required for the stability of the projection
approach, thus allowing to fully exploit the potential of higher order methods,
which makes the resulting approach an effective alternative to the use of the
finite element method