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 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.

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 A global div-curl-lemma for mixed boundary conditions in weak Lipschitz domains and a corresponding generalized A 0 * \mathrm{A}_{0}^{*} - A 1 \mathrm{A}_{1} -lemma in Hilbert spaces

Analysis ◽  
2019 ◽  
Vol 39 (2) ◽  
pp. 33-58 ◽  
Author(s):  
Dirk Pauly
Keyword(s):  
Boundary Conditions ◽  
Hilbert Spaces ◽  
Differential Operators ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Homogenization Theory ◽  
Lipschitz Domains ◽  
Compact Embeddings ◽  
Arbitrary Dimensions ◽  
Global And Local

Abstract We prove global and local versions of the so-called {\operatorname{div}} - {\operatorname{curl}} -lemma, a crucial result in the homogenization theory of partial differential equations, for mixed boundary conditions on bounded weak Lipschitz domains in 3D with weak Lipschitz interfaces. We will generalize our results using an abstract Hilbert space setting, which shows corresponding results to hold in arbitrary dimensions as well as for various differential operators. The crucial tools and the core of our arguments are Hilbert complexes and related compact embeddings.

Upon Impact Numerical Modeling of Foam Materials

2017 ◽  
Vol 54 (2) ◽  
pp. 195-202
Author(s):  
Vasile Nastasescu ◽  
Silvia Marzavan
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Modeling ◽  
Numerical Analysis ◽  
Numerical Methods ◽  
Boundary Conditions ◽  
General Character ◽  
Foam Material ◽  
Various Boundary Conditions ◽  
Specific Behaviour

The paper presents some theoretical and practical issues, particularly useful to users of numerical methods, especially finite element method for the behaviour modelling of the foam materials. Given the characteristics of specific behaviour of the foam materials, the requirement which has to be taken into consideration is the compression, inclusive impact with bodies more rigid then a foam material, when this is used alone or in combination with other materials in the form of composite laminated with various boundary conditions. The results and conclusions presented in this paper are the results of our investigations in the field and relates to the use of LS-Dyna program, but many observations, findings and conclusions, have a general character, valid for use of any numerical analysis by FEM programs.

Spectral Methods

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Simple Model ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Linear Algebra ◽  
Spectral Methods ◽  
Differential Operators ◽  
Higher Dimensions ◽  
Standard Linear ◽  
Sommerfeld Equation ◽  
Spectral Discretization

Thenumerical solution of ordinary differential equations (ODEs)with boundary conditions is studied here. Functions are approximated by polynomials in a Chebychev basis. Sections then cover spectral discretization, sampling, interpolation, differentiation, integration, and the basic ODE. Following Trefethen et al., differential operators are approximated as rectangular matrices. Boundary conditions add additional rows that turn them into square matrices. These can then be diagonalized using standard linear algebra methods. After studying various simple model problems, this method is applied to the Orr–Sommerfeld equation, deriving results originally due to Orszag. The difficulties of pushing spectral methods to higher dimensions are outlined.

Sign in / Sign up

Export Citation Format

Share Document