12 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 solver for conforming multipatch Isogeometric Analysis
In this paper we present a low-rank method for conforming multipatch
discretizations of compressible linear elasticity problems using Isogeometric
Analysis. The proposed technique is a non-trivial extension of [M. Montardini,
G. Sangalli, and M. Tani. A low-rank isogeometric solver based on Tucker
tensors. Comput. Methods Appl. Mech. Engrg., page 116472, 2023.] to multipatch
geometries. We tackle the model problem using an overlapping Schwarz method,
where the subdomains can be defined as unions of neighbouring patches. Then on
each subdomain we approximate the blocks of the linear system matrix and of the
right-hand side vector using Tucker matrices and Tucker vectors, respectively.
We use the Truncated Preconditioned Conjugate Gradient as a linear solver,
coupled with a suited preconditioner. The numerical experiments show the
advantages of this approach in terms of memory storage. Moreover, the number of
iterations is robust with respect to the relevant parameters.Comment: 17 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
Parallelization in time by diagonalization
This is a review of preconditioning techniques based on fast-diagonalization
methods for space-time isogeometric discretization of the heat equation. Three
formulation are considered: the Galerkin approach, a discrete least-square and
a continuous least square. For each formulation the heat differential operator
is written as a sum of terms that are kronecker products of uni-variate
operators. These are used to speed-up the application of the operator in
iterative solvers and to construct a suitable preconditioner. Contrary to the
fast-diagonalization technique for the Laplace equation where all uni-variate
operators acting on the same direction can be simultaneously diagonalized in
the case of the heat equation this is not possible. Luckily this can be done up
to an additional term that has low rank allowing for the utilization of
arrow-head like factorization or inversion by Sherman-Morrison formula. The
proposed preconditioners work extremely well on the parametric domain and, when
the domain is parametrized or when the equation coefficients are not constant,
they can be adapted and retain good performance characteristics.Comment: arXiv admin note: substantial text overlap with arXiv:1909.07309,
arXiv:2311.1846
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