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.
We propose an \emph{almost-robust} residual-based a-posteriori estimator for the advection-diffusion-reaction model problem. The theory is developed in the one-dimensional setting. The numerical ...error is measured with respect to a norm which was introduced by the author in 2005 and somehow plays the role that the \emph{energy} norm has with respect to symmetric and coercive differential operators. In particular, the mentioned norm possesses features that allow us to obtain a meaningful a-posteriori estimator, robust up to a \sqrt{\log(Pe)} factor, where Pe is the global \emph{Péclet number} of the problem. Various numerical tests are performed in one dimension, to confirm the theoretical results and show that the proposed estimator performs better than the usual one known in literature. We also consider a possible two-dimensional extension of our result and only present a few basic numerical tests, indicating that the estimator seems to preserve the good features of the one-dimensional setting.
A family of C1 quadrilateral finite elements Kapl, Mario; Sangalli, Giancarlo; Takacs, Thomas
Advances in computational mathematics,
2021/12, Letnik:
47, Številka:
6
Journal Article
Recenzirano
Odprti dostop
We present a novel family of
C
1
quadrilateral finite elements, which define global
C
1
spaces over a general quadrilateral mesh with vertices of arbitrary valency. The elements extend the ...construction by Brenner and Sung (J. Sci. Comput. 22(1-3), 83-118,
2005
), which is based on polynomial elements of tensor-product degree
p
≥ 6, to all degrees
p
≥ 3. The proposed
C
1
quadrilateral is based upon the construction of multi-patch
C
1
isogeometric spaces developed in Kapl et al. (Comput. Aided Geometr. Des.
69
, 55–75
2019
). The quadrilateral elements possess similar degrees of freedom as the classical Argyris triangles, developed in Argyris et al. (Aeronaut. J.
72
(692), 701–709
1968
). Just as for the Argyris triangle, we additionally impose
C
2
continuity at the vertices. In contrast to Kapl et al. (Comput. Aided Geometr. Des.
69
, 55–75
2019
), in this paper, we concentrate on quadrilateral finite elements, which significantly simplifies the construction. We present macro-element constructions, extending the elements in Brenner and Sung (J. Sci. Comput.
22
(1–3), 83–118
2005
), for polynomial degrees
p
= 3 and
p
= 4 by employing a splitting into 3 × 3 or 2 × 2 polynomial pieces, respectively. We moreover provide approximation error bounds in
L
∞
,
L
2
,
H
1
and
H
2
for the piecewise-polynomial macro-element constructions of degree
p
∈{3,4} and polynomial elements of degree
p
≥ 5. Since the elements locally reproduce polynomials of total degree
p
, the approximation orders are optimal with respect to the mesh size. Note that the proposed construction combines the possibility for spline refinement (equivalent to a regular splitting of quadrilateral finite elements) as in Kapl et al. (Comput. Aided Geometr. Des.
69
, 55–75
30
) with the purely local description of the finite element space and basis as in Brenner and Sung (J. Sci. Comput.
22
(1–3), 83–118
2005
). In addition, we describe the construction of a simple, local basis and give for
p
∈{3,4,5} explicit formulas for the Bézier or B-spline coefficients of the basis functions. Numerical experiments by solving the biharmonic equation demonstrate the potential of the proposed
C
1
quadrilateral finite element for the numerical analysis of fourth order problems, also indicating that (for
p
= 5) the proposed element performs comparable or in general even better than the Argyris triangle with respect to the number of degrees of freedom.
We propose an innovative isogeometric space–time method for the heat equation, with smooth splines approximation in both space and time. To enhance the stability of the method we add a stabilizing ...term, based on a linear combination of high-order artificial diffusions. This term is designed in order to make the linear system lower block-triangular, that is, lower triangular with respect to time. In order to keep optimal accuracy, the stabilization terms are further weighted in terms of the residual. Through a series of numerical experiments, we validate the method’s capability, showcasing its stability and accuracy.
We propose new collocation methods for phase-field models. Our algorithms are based on isogeometric analysis, a new technology that makes use of functions from computational geometry, such as, for ...example, Non-Uniform Rational B-Splines (NURBS). NURBS exhibit excellent approximability and controllable global smoothness, and can represent exactly most geometries encapsulated in Computer Aided Design (CAD) models. These attributes permitted us to derive accurate, efficient, and geometrically flexible collocation methods for phase-field models. The performance of our method is demonstrated by several numerical examples of phase separation modeled by the Cahn–Hilliard equation. We feel that our method successfully combines the geometrical flexibility of finite elements with the accuracy and simplicity of pseudo-spectral collocation methods, and is a viable alternative to classical collocation methods.
This paper deals with the fast solution of linear systems associated with the mass matrix, in the context of isogeometric analysis. We propose a preconditioner that is both efficient and easy to ...implement, based on a diagonal-scaled Kronecker product of univariate parametric mass matrices. Its application is faster than a matrix–vector product involving the mass matrix itself. We prove that the condition number of the preconditioned matrix converges to 1 as the mesh size is reduced, that is, the preconditioner is asymptotically equivalent to the exact inverse. Moreover, we give numerical evidence of its good behaviour with respect to the spline degree and the (possibly singular) geometry parametrization. We also extend the preconditioner to the multipatch case through an Additive Schwarz method.
In this paper we propose a Lagrange multiplier method for the finite element solution of multi-domain elliptic partial differential equations using non-matching meshes. The interface Lagrange ...multiplier is chosen with the purpose of avoiding the cumbersome integration of products of functions on unrelated meshes (e.g, we will consider global polynomials as multiplier). The ideas are illustrated using Poissons equation as a model, and the proposed method is shown to be stable and optimally convergent. Numerical experiments demonstrating the theoretical results are also presented.
This work introduces a novel isogeometric method for addressing steady, saturated groundwater flow as a free-boundary problem. The innovation lies in adopting a double spline parameterization ...technique for accurately representing both the porous medium and the fully saturated zone. To linearise the governing equation, we differentiate them with respect to the coefficients of the geometric parameterization. This process results in a quasi-Newton method. Several benchmark numerical tests demonstrate the applicability and effectiveness of the proposed technique.
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 quasi-robustness with respect to the local mesh-size h and patch-size H: precisely the condition number of the preconditioned system is bounded by the square of the logarithm of H∕h. Our numerical tests confirm the theory and also show a favourable dependence of the computational cost of the method from the spline degree p.
A sparse-grid isogeometric solver Beck, Joakim; Sangalli, Giancarlo; Tamellini, Lorenzo
Computer methods in applied mechanics and engineering,
06/2018, Letnik:
335
Journal Article
Recenzirano
Odprti dostop
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.
•A sparse grid version of classical isogeometric solvers is proposed.•It is competitive with standard solvers if the PDE solution is smooth.•It can reuse pre-existing solvers and make them parallel in a “black-box” way.•Radical meshes can be adopted to improve convergence for non-smooth solution.
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