The Construction of the Coarse de Rham Complexes with Improved Approximation Properties

2014 ◽  
Vol 14 (2) ◽  
pp. 257-303 ◽  
Author(s):  
Ilya V. Lashuk ◽  
Panayot S. Vassilevski
Keyword(s):  
Numerical Experiments ◽  
Differential Operators ◽  
Piecewise Linear ◽  
Tetrahedral Mesh ◽  
Approximation Properties ◽  
Fine Grid ◽  
De Rham ◽  
Affine Functions ◽  
Unstructured Tetrahedral Meshes ◽  
Coarse Space

Abstract. We present two novel coarse spaces (H1- and $H(\operatorname{curl})$-conforming) based on element agglomeration on unstructured tetrahedral meshes. Each H1-conforming coarse basis function is continuous and piecewise-linear with respect to an original tetrahedral mesh. The $H(\operatorname{curl})$-conforming coarse space is a subspace of the lowest order Nédélec space of the first type. The H1-conforming coarse space exactly interpolates affine functions on each agglomerate. The $H(\operatorname{curl})$-conforming coarse space exactly interpolates vector constants on each agglomerate. Combined with the $H(\operatorname{div})$- and L2-conforming spaces developed previously in [Numer. Linear Algebra Appl. 19 (2012), 414–426], the newly constructed coarse spaces form a sequence (with respect to exterior derivatives) which is exact as long as the underlying sequence of fine-grid spaces is exact. The constructed coarse spaces inherit the approximation and stability properties of the underlying fine-grid spaces supported by our numerical experiments. The new coarse spaces, in addition to multigrid, can be used for upscaling of broad range of PDEs involving $\operatorname{curl}$, $\operatorname{div}$ and $\operatorname{grad}$ differential operators.

scholarly journals Mollification in Strongly Lipschitz Domains with Application to Continuous and Discrete De Rham Complexes

2016 ◽  
Vol 16 (1) ◽  
pp. 51-75 ◽  
Author(s):  
Alexandre Ern ◽  
Jean-Luc Guermond
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Vector Fields ◽  
Differential Operators ◽  
Lipschitz Domains ◽  
Projection Operators ◽  
Approximation Properties ◽  
Smooth Functions ◽  
Conforming Finite Element ◽  
De Rham

AbstractWe construct mollification operators in strongly Lipschitz domains that do not invoke non-trivial extensions, are Lp stable for any real number ${p\in [1,\infty ]}$, and commute with the differential operators ∇, ${\nabla {\times }}$, and ${\nabla {\cdot }}$. We also construct mollification operators satisfying boundary conditions and use them to characterize the kernel of traces related to the tangential and normal trace of vector fields. We use the mollification operators to build projection operators onto general H1-, ${{H}(\mathrm {curl})}$- and ${{H}(\mathrm {div})}$-conforming finite element spaces, with and without homogeneous boundary conditions. These operators commute with the differential operators ∇, ${\nabla {\times }}$, and ${\nabla {\cdot }}$, are Lp-stable, and have optimal approximation properties on smooth functions.

scholarly journals Finite element algorithms for nonlocal minimal graphs

2021 ◽  
Vol 4 (2) ◽  
pp. 1-29
Author(s):  
Juan Pablo Borthagaray ◽  
◽  
Wenbo Li ◽  
Ricardo H. Nochetto ◽  
◽  
...  
Keyword(s):  
Mean Curvature ◽  
Numerical Experiments ◽  
Piecewise Linear ◽  
Gradient Flow ◽  
Minimal Graphs ◽  
Qualitative And Quantitative ◽  
Dirichlet Data ◽  
Data Truncation ◽  
Discrete Problems ◽  
Newton Scheme

<abstract><p>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 minimization, and we discretize the latter with piecewise linear finite elements. For the computation of the discrete solutions, we propose and study a gradient flow and a Newton scheme, and we quantify the effect of Dirichlet data truncation. We also present a wide variety of numerical experiments that illustrate qualitative and quantitative features of fractional minimal graphs and the associated discrete problems.</p></abstract>

Nonlocality of one-dimensional bilinear hardening–softening elastoplastic axial lattices

2019 ◽  
Vol 25 (2) ◽  
pp. 475-497
Author(s):  
Vincent Picandet ◽  
Noël Challamel
Keyword(s):  
Difference Equations ◽  
Differential Operators ◽  
Scale Effects ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Continuous Model ◽  
Lattice System ◽  
Discrete Problem ◽  
Linear Difference Equations ◽  
One Dimensional

The static behaviour of an elastoplastic axial lattice is studied in this paper through both discrete and nonlocal continuum analyses. The elastoplastic lattice system is composed of piecewise linear hardening–softening elastoplastic springs connected between each other via nodes, loaded by concentrated tension forces. This inelastic lattice evolution problem is ruled by some difference equations, which are shown to be equivalent to the finite difference formulation of a continuous elastoplastic bar problem under distributed tension load. Exact solutions of this inelastic discrete problem are obtained from the resolution of this piecewise linear difference equations system. Localization of plastic strain in the first elastoplastic spring, connected to the fixed end, is observed in the softening range. A continuous nonlocal elastoplastic theory is then built from the lattice difference equations using a continualization process, based on a rational asymptotic expansion of the associated pseudo-differential operators. The continualized lattice-based model is equivalent to a distributed nonlocal continuous elastoplastic theory coupled to a cohesive elastoplastic model, which is shown to capture efficiently the scale effects of the reference axial lattice. The hardening–softening localization process of the nonlocal elastoplastic continuous model strongly depends on the lattice spacing, which controls the size of the nonlocal length scales. An analogy with the one-dimensional lattice system in bending is also shown. The considered microstructured elastoplastic beam is a Hencky bar-chain connected by elastoplastic rotational springs. It is shown that the length scale calibration of the nonlocal model strongly depends on the degree of the difference equations of each lattice model (namely axial or bending lattice). These preliminary results valid for one-dimensional systems allow possible future developments of new nonlocal elastoplastic models, including two- or even three-dimensional elastoplastic interactions.

scholarly journals Sharp eigenvalue enclosures for the perturbed angular Kerr–Newman Dirac operator

2015 ◽  
Vol 471 (2182) ◽  
pp. 20150232
Author(s):  
Lyonell Boulton ◽  
Monika Winklmeier
Keyword(s):  
Dirac Operator ◽  
Differential Operators ◽  
Piecewise Linear ◽  
Rates Of Convergence ◽  
Linear Functions ◽  
Piecewise Linear Functions ◽  
Numerical Tests ◽  
Matrix Differential Operators ◽  
Singular Coefficients ◽  
Matrix Differential

We examine a certified strategy for determining sharp intervals of enclosure for the eigenvalues of matrix differential operators with singular coefficients. The strategy relies on computing the second-order spectrum relative to subspaces of continuous piecewise linear functions. For smooth perturbations of the angular Kerr–Newman Dirac operator, explicit rates of convergence linked to regularity of the eigenfunctions are established. Numerical tests which validate and sharpen by several orders of magnitude the existing benchmarks are also included.

scholarly journals A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Chuanjun Chen ◽  
Wei Liu
Keyword(s):  
Finite Element ◽  
Piecewise Linear ◽  
A Priori ◽  
Grid Method ◽  
Finite Element Approximation ◽  
Nonlinear Parabolic Equations ◽  
Coarse Grid ◽  
Element Approximation ◽  
Fine Grid ◽  
Nonlinear Parabolic

A two-grid method is presented and discussed for a finite element approximation to a nonlinear parabolic equation in two space dimensions. Piecewise linear trial functions are used. In this two-grid scheme, the full nonlinear problem is solved only on a coarse grid with grid sizeH. The nonlinearities are expanded about the coarse grid solution on a fine gird of sizeh, and the resulting linear system is solved on the fine grid. A priori error estimates are derived with theH1-normO(h+H2)which shows that the two-grid method achieves asymptotically optimal approximation as long as the mesh sizes satisfyh=O(H2). An example is also given to illustrate the theoretical results.

scholarly journals Differential Calculus onN-Graded Manifolds

2017 ◽  
Vol 2017 ◽  
pp. 1-19
Author(s):  
G. Sardanashvily ◽  
W. Wachowski
Keyword(s):  
Differential Calculus ◽  
Differential Operators ◽  
Differential Forms ◽  
Commutative Rings ◽  
Rham Complex ◽  
Grassmann Algebras ◽  
De Rham ◽  
Linear Differential ◽  
Graded Manifold ◽  
Graded Manifolds

The differential calculus, including formalism of linear differential operators and the Chevalley–Eilenberg differential calculus, overN-graded commutative rings and onN-graded manifolds is developed. This is a straightforward generalization of the conventional differential calculus over commutative rings and also is the case of the differential calculus over Grassmann algebras and onZ2-graded manifolds. We follow the notion of anN-graded manifold as a local-ringed space whose body is a smooth manifoldZ. A key point is that the graded derivation module of the structure ring of graded functions on anN-graded manifold is the structure ring of global sections of a certain smooth vector bundle over its bodyZ. Accordingly, the Chevalley–Eilenberg differential calculus on anN-graded manifold provides it with the de Rham complex of graded differential forms. This fact enables us to extend the differential calculus onN-graded manifolds to formalism of nonlinear differential operators, by analogy with that on smooth manifolds, in terms of graded jet manifolds ofN-graded bundles.

RANK ONE CHAOS IN SWITCH-CONTROLLED PIECEWISE LINEAR CHUA'S CIRCUIT

2007 ◽  
Vol 16 (05) ◽  
pp. 769-789 ◽  
Author(s):  
QIUDONG WANG ◽  
ALI OKSASOGLU
Keyword(s):  
Periodic Solution ◽  
Numerical Experiments ◽  
Piecewise Linear ◽  
Small Neighborhood ◽  
Hopf Bifurcations ◽  
Asymptotically Stable ◽  
Stable Periodic Solution ◽  
Pspice Simulations ◽  
Rank One ◽  
Stable Periodic

In this paper, we continue our study of rank one chaos in switch-controlled circuits. Periodically controlled switches are added to Chua's original piecewise linear circuit to generate rank one attractors in the vicinity of an asymptotically stable periodic solution that is relatively large in size. Our previous investigations relied heavily on the smooth nonlinearity of the unforced systems, and were, by large, restricted to a small neighborhood of supercritical Hopf bifurcations. Whereas the system studied in this paper is much more feasible for physical implementation, and thus the corresponding rank one chaos is much easier to detect in practice. The findings of our purely numerical experiments are further supported by the PSPICE simulations.

scholarly journals Fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra

2020 ◽  
Vol 30 (09) ◽  
pp. 1809-1855
Author(s):  
Daniele A. Di Pietro ◽  
Jérôme Droniou ◽  
Francesca Rapetti
Keyword(s):  
Finite Element ◽  
Commutative Diagram ◽  
Three Dimensional ◽  
Computer Implementation ◽  
Approximation Properties ◽  
Arbitrary Degree ◽  
Fully Discrete ◽  
De Rham ◽  
Interpolation Operators ◽  
Suitable Approximation

In this work, merging ideas from compatible discretisations and polyhedral methods, we construct novel fully discrete polynomial de Rham sequences of arbitrary degree on polygons and polyhedra. The spaces and operators that appear in these sequences are directly amenable to computer implementation. Besides proving the exactness, we show that the usual three-dimensional sequence of trimmed Finite Element (FE) spaces forms, through appropriate interpolation operators, a commutative diagram with our sequence, which ensures suitable approximation properties. A discussion on reconstructions of potentials and discrete [Formula: see text]-products completes the exposition.

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