109 research outputs found
Finite element differential forms on curvilinear cubic meshes and their approximation properties
We study the approximation properties of a wide class of finite element
differential forms on curvilinear cubic meshes in n dimensions. Specifically,
we consider meshes in which each element is the image of a cubical reference
element under a diffeomorphism, and finite element spaces in which the shape
functions and degrees of freedom are obtained from the reference element by
pullback of differential forms. In the case where the diffeomorphisms from the
reference element are all affine, i.e., mesh consists of parallelotopes, it is
standard that the rate of convergence in L2 exceeds by one the degree of the
largest full polynomial space contained in the reference space of shape
functions. When the diffeomorphism is multilinear, the rate of convergence for
the same space of reference shape function may degrade severely, the more so
when the form degree is larger. The main result of the paper gives a sufficient
condition on the reference shape functions to obtain a given rate of
convergence.Comment: 17 pages, 1 figure; v2: changes in response to referee reports; v3:
minor additional changes, this version accepted for Numerische Mathematik;
v3: very minor updates, this version corresponds to the final published
versio
Regularity and sparse approximation of the recursive first moment equations for the lognormal Darcy problem
We study the Darcy boundary value problem with log-normal permeability field.
We adopt a perturbation approach, expanding the solution in Taylor series
around the nominal value of the coefficient, and approximating the expected
value of the stochastic solution of the PDE by the expected value of its Taylor
polynomial. The recursive deterministic equation satisfied by the expected
value of the Taylor polynomial (first moment equation) is formally derived.
Well-posedness and regularity results for the recursion are proved to hold in
Sobolev space-valued H\"older spaces with mixed regularity. The recursive first
moment equation is then discretized by means of a sparse approximation
technique, and the convergence rates are derived
H1-conforming finite element cochain complexes and commuting quasi-interpolation operators on cartesian meshes
A finite element cochain complex on Cartesian meshes of any dimension based
on the H1-inner product is introduced. It yields H1-conforming finite element
spaces with exterior derivatives in H1. We use a tensor product construction to
obtain L2-stable projectors into these spaces which commute with the exterior
derivative. The finite element complex is generalized to a family of arbitrary
order
A DG-VEM method for the dissipative wave equation
A novel space-time discretization for the (linear) scalar-valued dissipative
wave equation is presented. It is a structured approach, namely, the
discretization space is obtained tensorizing the Virtual Element (VE)
discretization in space with the Discontinuous Galerkin (DG) method in time. As
such, it combines the advantages of both the VE and the DG methods. The
proposed scheme is implicit and it is proved to be unconditionally stable and
accurate in space and time
Moment equations for the mixed formulation of the Hodge Laplacian with stochastic loading term
We study the mixed formulation of the stochastic Hodge-Laplace problem defined on an n-dimensional domain D (n≥1), with random forcing term. In particular, we focus on the magnetostatic problem and on the Darcy problem in the three-dimensional case. We derive and analyse the moment equations, that is, the deterministic equations solved by the mth moment (m≥1) of the unique stochastic solution of the stochastic problem. We find stable tensor product finite element discretizations, both full and sparse, and provide optimal order-of-convergence estimates. In particular, we prove the inf-sup condition for sparse tensor product finite element space
Uncertainty quantification in timber-like beams using sparse grids: theory and examples with off-the-shelf software utilization
When dealing with timber structures, the characteristic strength and
stiffness of the material are made highly variable and uncertain by the
unavoidable, yet hardly predictable, presence of knots and other defects. In
this work we apply the sparse grids stochastic collocation method to perform
uncertainty quantification for structural engineering in the scenario described
above. Sparse grids have been developed by the mathematical community in the
last decades and their theoretical background has been rigorously and
extensively studied. The document proposes a brief practice-oriented
introduction with minimal theoretical background, provides detailed
instructions for the use of the already implemented Sparse Grid Matlab kit
(freely available on-line) and discusses two numerical examples inspired from
timber engineering problems that highlight how sparse grids exhibit superior
performances compared to the plain Monte Carlo method. The Sparse Grid Matlab
kit requires only a few lines of code to be interfaced with any numerical
solver for mechanical problems (in this work we used an isogeometric
collocation method) and provides outputs that can be easily interpreted and
used in the engineering practice
A reduced basis super-localized orthogonal decomposition for reaction-convection-diffusion problems
This paper presents a method for the numerical treatment of
reaction-convection-diffusion problems with parameter-dependent coefficients
that are arbitrary rough and possibly varying at a very fine scale. The
presented technique combines the reduced basis (RB) framework with the recently
proposed super-localized orthogonal decomposition (SLOD). More specifically,
the RB is used for accelerating the typically costly SLOD basis computation,
while the SLOD is employed for an efficient compression of the problem's
solution operator requiring coarse solves only. The combined advantages of both
methods allow one to tackle the challenges arising from parametric
heterogeneous coefficients. Given a value of the parameter vector, the method
outputs a corresponding compressed solution operator which can be used to
efficiently treat multiple, possibly non-affine, right-hand sides at the same
time, requiring only one coarse solve per right-hand side.Comment: 27 pages, 6 figure
A cVEM-DG space-time method for the dissipative wave equation
A novel space-time discretization for the (linear) scalar-valued dissipative wave equation is presented. It is a structured approach, namely, the discretization space is obtained tensorizing the Virtual Element (VE) discretization in space with the Discontinuous Galerkin (DG) method in time. As such, it combines the advantages of both the VE and the DG methods. The proposed scheme is implicit and it is proved to be unconditionally stable and accurate in space and time
Fast Least-Squares Pad\'e approximation of problems with normal operators and meromorphic structure
In this work, we consider the approximation of Hilbert space-valued
meromorphic functions that arise as solution maps of parametric PDEs whose
operator is the shift of an operator with normal and compact resolvent, e.g.
the Helmholtz equation. In this restrictive setting, we propose a simplified
version of the Least-Squares Pad\'e approximation technique introduced in [6]
following [11]. In particular, the estimation of the poles of the target
function reduces to a low-dimensional eigenproblem for a Gramian matrix,
allowing for a robust and efficient numerical implementation (hence the "fast"
in the name). Moreover, we prove several theoretical results that improve and
extend those in [6], including the exponential decay of the error in the
approximation of the poles, and the convergence in measure of the approximant
to the target function. The latter result extends the classical one for scalar
Pad\'e approximation to our functional framework. We provide numerical results
that confirm the improved accuracy of the proposed method with respect to the
one introduced in [6] for differential operators with normal and compact
resolvent
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