2,066 research outputs found
A robust a posteriori estimator for the residual-free bubbles method applied to advection-diffusion problems
We develop the a posteriori error analysis for the RFB method, applied to the linear advection-diffusion problem: the numerical error, measured in suitable norms, is estimated in terms of the numerical residual. The robustness is investiged, in the sense that we prove uniform equivalence between a norm of the numerical residual and a particular norm of the error
Isogeometric preconditioners based on fast solvers for the Sylvester equation
We consider large linear systems arising from the isogeometric discretization
of the Poisson problem on a single-patch domain. The numerical solution of such
systems is considered a challenging task, particularly when the degree of the
splines employed as basis functions is high. We consider a preconditioning
strategy which is based on the solution of a Sylvester-like equation at each
step of an iterative solver. We show that this strategy, which fully exploits
the tensor structure that underlies isogeometric problems, is robust with
respect to both mesh size and spline degree, although it may suffer from the
presence of complicated geometry or coefficients. We consider two popular
solvers for the Sylvester equation, a direct one and an iterative one, and we
discuss in detail their implementation and efficiency for 2D and 3D problems on
single-patch or conforming multi-patch NURBS geometries. Numerical experiments
for problems with different domain geometries are presented, which demonstrate
the potential of this approach
Matrix-free weighted quadrature for a computationally efficient isogeometric -method
The -method is the isogeometric method based on splines (or NURBS, etc.)
with maximum regularity. When implemented following the paradigms of classical
finite element methods, the computational resources required by the method
are prohibitive even for moderate degree. In order to address this issue, we
propose a matrix-free strategy combined with weighted quadrature, which is an
ad-hoc strategy to compute the integrals of the Galerkin system. Matrix-free
weighted quadrature (MF-WQ) speeds up matrix operations, and, perhaps even more
important, greatly reduces memory consumption. Our strategy also requires an
efficient preconditioner for the linear system iterative solver. In this work
we deal with an elliptic model problem, and adopt a preconditioner based on the
Fast Diagonalization method, an old idea to solve Sylvester-like equations. Our
numerical tests show that the isogeometric solver based on MF-WQ is faster than
standard approaches (where the main cost is the matrix formation by standard
Gaussian quadrature) even for low degree. But the main achievement is that,
with MF-WQ, the -method gets orders of magnitude faster by increasing the
degree, given a target accuracy. Therefore, we are able to show the
superiority, in terms of computational efficiency, of the high-degree
-method with respect to low-degree isogeometric discretizations. What we
present here is applicable to more complex and realistic differential problems,
but its effectiveness will depend on the preconditioner stage, which is as
always problem-dependent. This situation is typical of modern high-order
methods: the overall performance is mainly related to the quality of the
preconditioner
Ultra--fast carriers relaxation in bulk silicon following photo--excitation with a short and polarized laser pulse
A novel approach based on the merging of the out--of--equilibrium Green's
function method with the ab-initio, Density--Functional--Theory is used to
describe the ultra--fast carriers relaxation in Silicon. The results are
compared with recent two photon photo--emission measurements. We show that the
interpretation of the carrier relaxation in terms of L -> X inter--valley
scattering is not correct. The ultra--fast dynamics measured experimentally is,
instead, due to the scattering between degenerate states that is activated
by the non symmetric population of the conduction bands induced by the laser
field. This ultra--fast relaxation is, then, entirely due to the specific
experimental setup and it can be interpreted by introducing a novel definition
of the quasi--particle lifetimes in an out--of--equilibrium context.Comment: 4 page, 2 figure
Generalized Spatial Regression with Differential Regularization
We aim at analyzing geostatistical and areal data observed over irregularly
shaped spatial domains and having a distribution within the exponential family.
We propose a generalized additive model that allows to account for
spatially-varying covariate information. The model is fitted by maximizing a
penalized log-likelihood function, with a roughness penalty term that involves
a differential quantity of the spatial field, computed over the domain of
interest. Efficient estimation of the spatial field is achieved resorting to
the finite element method, which provides a basis for piecewise polynomial
surfaces. The proposed model is illustrated by an application to the study of
criminality in the city of Portland, Oregon, USA
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
Construction of analysis-suitable planar multi-patch parameterizations
Isogeometric analysis allows to define shape functions of global
continuity (or of higher continuity) over multi-patch geometries. The
construction of such -smooth isogeometric functions is a non-trivial
task and requires particular multi-patch parameterizations, so-called
analysis-suitable (in short, AS-) parameterizations, to ensure
that the resulting isogeometric spaces possess optimal approximation
properties, cf. [7]. In this work, we show through examples that it is possible
to construct AS- multi-patch parameterizations of planar domains, given
their boundary. More precisely, given a generic multi-patch geometry, we
generate an AS- multi-patch parameterization possessing the same
boundary, the same vertices and the same first derivatives at the vertices, and
which is as close as possible to this initial geometry. Our algorithm is based
on a quadratic optimization problem with linear side constraints. Numerical
tests also confirm that isogeometric spaces over AS- multi-patch
parameterized domains converge optimally under mesh refinement, while for
generic parameterizations the convergence order is severely reduced
A sparse-grid isogeometric solver
Isogeometric Analysis (IGA) typically adopts tensor-product splines and NURBS
as a basis for the approximation of the solution of PDEs. In this work, we
investigate to which extent IGA solvers can benefit from the so-called
sparse-grids construction in its combination technique form, which was first
introduced in the early 90s in the context of the approximation of
high-dimensional PDEs. The tests that we report show that, in accordance to the
literature, a sparse-grid construction can indeed be useful if the solution of
the PDE at hand is sufficiently smooth. Sparse grids can also be useful in the
case of non-smooth solutions when some a-priori knowledge on the location of
the singularities of the solution can be exploited to devise suitable
non-equispaced meshes. Finally, we remark that sparse grids can be seen as a
simple way to parallelize pre-existing serial IGA solvers in a straightforward
fashion, which can be beneficial in many practical situations.Comment: updated version after revie
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