Ricardo H. Nochetto
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View article: Surface Stokes Without Inf-Sup Condition
Surface Stokes Without Inf-Sup Condition Open
For a $d$-dimensional hypersurface of class $C^3$ without boundary, we reformulate the surface Stokes equations as a nonsymmetric indefinite elliptic problem governed by two Laplacians. We then use this elliptic reformulation as a basis fo…
View article: Inf-Sup Stability of Parabolic TraceFEM
Inf-Sup Stability of Parabolic TraceFEM Open
We develop a parabolic inf-sup theory for a modified TraceFEM semi-discretization in space of the heat equation posed on a stationary surface embedded in $\mathbb{R}^n$. We consider the normal derivative volume stabilization and add an $L^…
View article: Adaptive finite element methods
Adaptive finite element methods Open
This is a survey of the theory of adaptive finite element methods (AFEMs), which are fundamental to modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and u…
View article: Monotone two-scale methods for a class of integrodifferential operators and applications
Monotone two-scale methods for a class of integrodifferential operators and applications Open
We develop a monotone, two-scale discretization for a class of integrodifferential operators of order $2s$, $s \in (0,1)$. We apply it to develop numerical schemes, and derive pointwise convergence rates, for linear and obstacle problems g…
View article: Projection-Free Method for the Full Frank-Oseen Model of Liquid Crystals
Projection-Free Method for the Full Frank-Oseen Model of Liquid Crystals Open
Liquid crystals are materials that experience an intermediate phase where the material can flow like a liquid, but the molecules maintain an orientation order. The Frank-Oseen model is a continuum model of a liquid crystal. The model repre…
View article: Conforming virtual element method for nondivergence form linear elliptic equations with Cordes coefficients
Conforming virtual element method for nondivergence form linear elliptic equations with Cordes coefficients Open
We propose and analyze an $H^2$-conforming Virtual Element Method (VEM) for the simplest linear elliptic PDEs in nondivergence form with Cordes coefficients. The VEM hinges on a hierarchical construction valid for any dimension $d \ge 2$. …
View article: Adaptive Finite Element Methods
Adaptive Finite Element Methods Open
This is a survey on the theory of adaptive finite element methods (AFEMs), which are fundamental in modern computational science and engineering but whose mathematical assessment is a formidable challenge. We present a self-contained and u…
View article: Robust BPX preconditioner for fractional Laplacians on bounded Lipschitz domains
Robust BPX preconditioner for fractional Laplacians on bounded Lipschitz domains Open
We propose and analyze a robust Bramble-Pasciak-Xu (BPX) preconditioner for the integral fractional Laplacian of order on bounded Lipschitz domains. Compared with the standard BPX preconditioner, an additional scaling factor , for some fi…
View article: Quasi-linear fractional-order operators in Lipschitz domains
Quasi-linear fractional-order operators in Lipschitz domains Open
We prove Besov boundary regularity for solutions of the homogeneous Dirichlet problem for fractional-order quasi-linear operators with variable coefficients on Lipschitz domains $Ω$ of $\mathbb{R}^d$. Our estimates are consistent with the …
View article: Adaptive VEM: Stabilization-Free A Posteriori Error Analysis and Contraction Property
Adaptive VEM: Stabilization-Free A Posteriori Error Analysis and Contraction Property Open
In the present paper we initiate the challenging task of building a mathematically sound theory for adaptive virtual element methods (AVEMs). Among the realm of polygonal meshes, we restrict our analysis to triangular meshes with hanging n…
View article: $Γ$-convergent LDG method for large bending deformations of bilayer plates
$Γ$-convergent LDG method for large bending deformations of bilayer plates Open
Bilayer plates are slender structures made of two thin layers of different materials. They react to environmental stimuli and undergo large bending deformations with relatively small actuation. The reduced model is a constrained minimizati…
View article: Fractional Elliptic Problems on Lipschitz Domains: Regularity and Approximation
Fractional Elliptic Problems on Lipschitz Domains: Regularity and Approximation Open
This survey hinges on the interplay between regularity and approximation for linear and quasi-linear fractional elliptic problems on Lipschitz domains. For the linear Dirichlet integral Laplacian, after briefly recalling Hölder regularity …
View article: Reduced Membrane Model for Liquid Crystal Polymer Networks: Asymptotics and Computation
Reduced Membrane Model for Liquid Crystal Polymer Networks: Asymptotics and Computation Open
We examine a reduced membrane model of liquid crystal polymer networks (LCNs) via asymptotics and computation. This model requires solving a minimization problem for a non-convex stretching energy. We show a formal asymptotic derivation of…
View article: Convergent FEM for a membrane model of liquid crystal polymer networks
Convergent FEM for a membrane model of liquid crystal polymer networks Open
We design a finite element method (FEM) for a membrane model of liquid crystal polymer networks (LCNs). This model consists of a minimization problem of a non-convex stretching energy. We discuss properties of this energy functional such a…
View article: A hydrodynamical model of nematic liquid crystal films with a general state of orientational order
A hydrodynamical model of nematic liquid crystal films with a general state of orientational order Open
We develop a Q-tensor model of nematic liquid crystals occupying a stationary surface which represents a fluidic material film in space. In addition to the evolution due to Landau--de\,Gennes energy the model includes a tangent viscous inc…
View article: Gamma-Convergent Projection-Free Finite Element Methods for Nematic Liquid Crystals: The Ericksen Model
Gamma-Convergent Projection-Free Finite Element Methods for Nematic Liquid Crystals: The Ericksen Model Open
The Ericksen model for nematic liquid crystals couples a director field with a scalar degree of orientation variable and allows the formation of various defects with finite energy. We propose a simple but novel finite element approximation…
View article: Besov regularity for the Dirichlet integral fractional Laplacian in Lipschitz domains
Besov regularity for the Dirichlet integral fractional Laplacian in Lipschitz domains Open
We prove Besov regularity estimates for the solution of the Dirichlet problem involving the integral fractional Laplacian of order $s$ in bounded Lipschitz domains $Ω$: \[ \begin{aligned} \|u\|_{\dot{B}^{s+r}_{2,\infty}(Ω)} \le C \|f\|_{L^…
View article: Constructive approximation on graded meshes for the integral fractional Laplacian
Constructive approximation on graded meshes for the integral fractional Laplacian Open
We consider the homogeneous Dirichlet problem for the integral fractional Laplacian $(-Δ)^s$. We prove optimal Sobolev regularity estimates in Lipschitz domains provided the solution is $C^s$ up to the boundary. We present the construction…
View article: Two-scale methods for convex envelopes
Two-scale methods for convex envelopes Open
We develop two-scale methods for computing the convex envelope of a continuous function over a convex domain in any dimension. This hinges on a fully nonlinear obstacle formulation (see A. M. Oberman [Proc. Amer. Math. Soc. 135 (2007), pp.…
View article: Equilibrium analysis of an immersed rigid leaflet by the virtual element method
Equilibrium analysis of an immersed rigid leaflet by the virtual element method Open
We study, both theoretically and numerically, the equilibrium of a hinged rigid leaflet with an attached rotational spring, immersed in a stationary incompressible fluid within a rigid channel. Through a careful investigation of the proper…
View article: Numerical analysis of the LDG method for large deformations of prestrained plates
Numerical analysis of the LDG method for large deformations of prestrained plates Open
A local discontinuous Galerkin (LDG) method for approximating large deformations of prestrained plates is introduced and tested on several insightful numerical examples in our previous computational work. This paper presents a numerical an…
View article: Gamma-convergent projection-free finite element methods for nematic liquid crystals: The Ericksen model
Gamma-convergent projection-free finite element methods for nematic liquid crystals: The Ericksen model Open
The Ericksen model for nematic liquid crystals couples a director field with a scalar degree of orientation variable, and allows the formation of various defects with finite energy. We propose a simple but novel finite element approximatio…
View article: Robust BPX preconditioner for the integral fractional Laplacian on bounded domains.
Robust BPX preconditioner for the integral fractional Laplacian on bounded domains. Open
We propose and analyze a robust BPX preconditioner for the integral fractional Laplacian on bounded domains. For either quasi-uniform grids or graded bisection grids, we show that the condition numbers of the resulting systems remain unifo…
View article: Local Energy Estimates for the Fractional Laplacian
Local Energy Estimates for the Fractional Laplacian Open
The integral fractional Laplacian of order $s \in (0,1)$ is a nonlocal operator. It is known that solutions to the Dirichlet problem involving such an operator exhibit an algebraic boundary singularity regardless of the domain regularity. …
View article: Finite element algorithms for nonlocal minimal graphs
Finite element algorithms for nonlocal minimal graphs Open
We discuss computational and qualitative aspects of the fractional Plateau and the prescribed fractional mean curvature problems on bounded domains subject to exterior data being a subgraph. We recast these problems in terms of energy mini…
View article: DG approach to large bending plate deformations with isometry constraint
DG approach to large bending plate deformations with isometry constraint Open
We propose a new discontinuous Galerkin (dG) method for a geometrically nonlinear Kirchhoff plate model for large isometric bending deformations. The minimization problem is nonconvex due to the isometry constraint. We present a practical …
View article: LDG approximation of large deformations of prestrained plates
LDG approximation of large deformations of prestrained plates Open
A reduced model for large deformations of prestrained plates consists of minimizing a second order bending energy subject to a nonconvex metric constraint. The former involves the second fundamental form of the middle plate and the later i…
View article: Discontinuous Galerkin approach to large bending deformation of a bilayer plate with isometry constraint
Discontinuous Galerkin approach to large bending deformation of a bilayer plate with isometry constraint Open
View article: Equilibrium analysis of an immersed rigid leaflet by the virtual element method
Equilibrium analysis of an immersed rigid leaflet by the virtual element method Open
We study, both theoretically and numerically, the equilibrium of a hinged rigid leaflet with an attached rotational spring, immersed in a stationary incompressible fluid within a rigid channel. Through a careful investigation of the proper…
View article: A structure-preserving FEM for the uniaxially constrained $$\mathbf{Q}$$-tensor model of nematic liquid crystals
A structure-preserving FEM for the uniaxially constrained $$\mathbf{Q}$$-tensor model of nematic liquid crystals Open