Poincaré Inequality for a Mesh-Dependent 2-Norm on Piecewise Linear Surfaces with Boundary

Shawn W. Walker

doi:10.1515/cmam-2020-0123

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Poincaré Inequality for a Mesh-Dependent 2-Norm on Piecewise Linear Surfaces with Boundary

Shawn W. Walker

Published/Copyright: June 23, 2021

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From the journal Computational Methods in Applied Mathematics Volume 22 Issue 1

Abstract

We establish several useful estimates for a non-conforming 2-norm posed on piecewise linear surface triangulations with boundary, with the main result being a Poincaré inequality. We also obtain equivalence of the non-conforming 2-norm posed on the true surface with the norm posed on a piecewise linear approximation. Moreover, we allow for free boundary conditions. The true surface is assumed to be $C^{2, 1}$ $C 2 , 1$ when free conditions are present; otherwise, $C^{2}$ $C 2$ is sufficient. The framework uses tools from differential geometry and the closest point map (see [G. Dziuk, Finite elements for the Beltrami operator on arbitrary surfaces, Partial Differential Equations and Calculus of Variations, Lecture Notes in Math. 1357, Springer, Berlin (1988), 142–155]) for approximating the full surface Hessian operator. We also present a novel way of applying the closest point map when dealing with surfaces with boundary. Connections with surface finite element methods for fourth-order problems are also noted.

Keywords: Surfaces With Boundary; Mesh-Dependent Norms; Non-Conforming Method; Surface Finite Elements

MSC 2010: 65N30; 35J40; 35Q72

A Differential Geometry

In this appendix, we review the differential geometry tools needed for working on manifolds [38, 25, 24, 16, 37]. Specifically, we review the basic notation of covariant, contravariant, and other differential geometry concepts.

A.1 Intrinsic

For the sake of generality, consider a 𝑑-dimensional Riemannian manifold $(Γ, g_{a b})$ $( Γ , g a ⁢ b )$ , where $g_{a b}$ $g a ⁢ b$ is the given metric tensor (discussed in Section A.1.2) defined over a (reference) domain $U ⊂ R d$ ; for simplicity of exposition, assume only one reference domain is needed to define the manifold (of course, this is not necessary). A point in 𝑈 is denoted by $( u 1 , u 2 , … , u d )$ ; in the special case of $d = 2$ that we are mainly concerned with, we may use $( u , v ) ∈ U$ . We refer to variables defined on 𝑈 as intrinsic quantities.

A.1.1 Tensor Index Notation

We use lower-case Greek indices ( $α , β , γ$ , etc.), which take values in ${ 1 , 2 , … , d }$ when referring to intrinsic variables. For example, $∂ α$ is the partial derivative with respect to the coordinate $u α$ for $α ∈ { 1 , 2 , … , d }$ . Covariant vectors are denoted with lower indices, e.g. $( v 1 , v 2 , … , v d )$ and contravariant vectors are denoted with upper indices, e.g. $( v 1 , v 2 , … , v d )$ . The 𝛽-th component of a covariant (contravariant) derivative is denoted by $∇ β$ ( $∇ β$ ).

Moreover, covariant and contravariant components of general tensor quantities use lower and upper Greek indices, respectively, e.g. $w α ⁢ β$ (covariant tensor), $w α ⁢ β$ (contravariant tensor), $w α β$ , $w α β$ (mixed tensor). We adopt the Einstein summation convention, i.e. repeated indices are summed over, e.g., $w α ⁢ r α ≡ ∑ α = 1 d w α ⁢ r α$ , where one index is lower and the other is upper. For example, it is not allowed to sum over two repeated lower indices. We use the Kronecker delta $δ α ⁢ β$ , $δ α ⁢ β$ , $δ α β$ , etc., with appropriate upper/lower indices depending on the context.

Furthermore, we use the letters 𝔞–𝔥 (with a different font for emphasis) as a non-numerical label to indicate a covariant, contravariant, or mixed tensor. For example, $v a$ refers to a covariant vector (not just a single component), i.e. $v a ≡ ( v 1 , … , v d )$ . Similarly, $∇ c ⁡ z = ( ∇ 1 ⁡ z , … , ∇ d ⁡ z )$ refers to a contravariant vector, where 𝑧 is a scalar quantity. For non-numerical labels, the specific symbol does not matter; it is simply a placeholder. When convenient, we use bold-face for vector and tensor quantities instead of writing out indices.

A.1.2 Main Concepts

The given metric $g a ⁢ b$ is a symmetric, covariant tensor with component functions $g α ⁢ β : U → R$ for $1 ≤ α , β ≤ d$ , which we assume are at least $C 1$ , and is uniformly positive definite. We write $g := det ⁡ g a ⁢ b$ , and the inverse metric tensor $g a ⁢ b$ is contravariant with components denoted $g α ⁢ β$ , where $g α ⁢ γ ⁢ g γ ⁢ β = δ α β$ . Note that $v a$ may be converted to $v b$ via $v β = g β ⁢ α ⁢ v α$ ; similarly, $w b$ may be converted to $w a$ by $w α = g α ⁢ β ⁢ w β$ . When convenient, we write $g a ⁢ b ≡ g = [ g α ⁢ β ] α , β = 1 2$ and $g a ⁢ b ≡ g - 1 = [ g α ⁢ β ] α , β = 1 2$ in standard matrix notation for the metric and inverse metric, respectively. Let $T 2 = T 2 ⁢ ( Γ )$ ( $T 2 = T 2 ⁢ ( Γ )$ ) be the set of covariant (contravariant) 2-tensors on Γ. Moreover, $S 2 ⊂ T 2$ and $S 2 ⊂ T 2$ are subsets of symmetric tensors, so then $g a ⁢ b ∈ S 2$ and $g a ⁢ b ∈ S 2$ .

The Christoffel symbols $Γ i ⁢ j k$ (of the second kind) are defined by

(A.1)

$Γ α ⁢ β γ := 1 2 ⁢ g μ ⁢ γ ⁢ ( ∂ α ⁡ g β ⁢ μ + ∂ β ⁡ g μ ⁢ α - ∂ μ ⁡ g α ⁢ β ) , 1 ≤ α , β , γ ≤ 2 ,$

where $Γ α ⁢ β γ = Γ β ⁢ α γ$ (see [25, 24]). With this, we recall the definition of covariant (contravariant) derivatives, denoted $∇ α$ ( $∇ α$ ), where 𝑓 is a scalar, $v b$ is a covariant vector, and $v c$ is a contravariant vector,

$∇ α ⁡ f = ∂ α ⁡ f ,$

$∇ α ⁡ ∇ β ⁡ f = ∂ α ⁡ ∂ β ⁡ f - ( ∂ γ ⁡ f ) ⁢ Γ α ⁢ β γ ,$

$∇ α ⁡ v β = ∂ α ⁡ v β - v γ ⁢ Γ β ⁢ α γ ,$

$∇ α ⁡ v γ = ∂ α ⁡ v γ + v β ⁢ Γ β ⁢ α γ , ∇ α ⁡ v α = ( g ) - 1 ⁢ ∂ α ⁡ ( v α ⁢ g ) .$

The metric satisfies (see [25]) $∇ γ ⁡ g α ⁢ β = 0$ , $∇ γ ⁡ g α ⁢ β = 0$ , $∇ γ ⁡ g = 0$ for $1 ≤ α , β , γ ≤ 2$ . The “area” element on the manifold Γ is denoted $d ⁢ S ⁢ ( g ) = g ⁢ d ⁢ u ≡ g ⁢ d ⁢ u 1 ⁢ ⋯ ⁢ d ⁢ u d$ , where $d ⁢ u$ is the Lebesgue measure on $R d$ . Viewing $n a$ as a “vector” in $R d$ , it has unit length under the $R d$ Euclidean metric. If $d = 2$ , let $t a$ be the oriented (contravariant) tangent vector of $∂ ⁡ U$ , which has unit length in the Euclidean metric and satisfies $n α ⁢ t α = 0$ . Moreover, $g = t μ ⁢ t μ / ( n μ ⁢ n μ )$ , which implies that $d ⁢ s ⁢ ( g ) := t μ ⁢ t μ ⁢ d ⁢ l$ for $d = 2$ , and we have the following “orthogonal” decomposition:

$δ β α = n α ⁢ n β n μ ⁢ n μ + t α ⁢ t β t μ ⁢ t μ .$

A.2 Extrinsic

Suppose that the manifold Γ is embedded in $R n$ , with $n ≥ d$ , and that it is represented by a family of charts ${ ( U i , χ i ) }$ , where a single chart consists of a pair $( U , χ )$ , with $U ⊂ R d$ (reference domain) and $χ : U → R n$ (see [25]). For simplicity of exposition, assume there is only one chart $( U , χ )$ , where $Γ = χ ⁢ ( U )$ . We refer to variables in $R n$ as extrinsic quantities.

A.2.1 Tensor Index Notation

We use lower-case Latin letters starting with 𝑖 (i.e. $i , j , k , l$ , etc.), which take values in ${ 1 , 2 , … , n }$ , when referring to components of extrinsic (ambient space) quantities. For example, $χ = ( χ 1 , … , χ n ) T ∈ R n$ , and $χ i : U → R$ for each $i ∈ { 1 , 2 , … , n }$ . A point $x ∈ R n$ has its 𝑗-th coordinate denoted by $x j$ . Moreover, $∂ k$ is the partial derivative with respect to coordinate $x k$ . Repeated indices are summed over. We typically bold-face extrinsic vectors and tensors, e.g. let 𝒘 be a (covariant) 2-tensor in $R n$ with components $w i ⁢ j$ for $i , j ∈ { 1 , 2 , … , n }$ . The canonical (orthonormal) basis in $R n$ is denoted by ${ a k } k = 1 n$ , where $a 1 = ( 1 , 0 , … , 0 ) T$ (column vector), etc. With the Kronecker delta $δ i j$ , we have the dual basis ${ a k }$ of ${ a k }$ by the formula $a i ⋅ a j = δ i j$ .

A.2.2 Differential Geometry in the Ambient Space

The tangent space $T x ⁢ ( Γ )$ , at a point $x ∈ Γ$ , is a subspace of $R n$ spanned by ${ e 1 , e 2 , … , e d }$ (the covariant basis), where

$e α = ∂ α ⁡ χ ⁢ ( u a ) , 1 ≤ α ≤ d , where ⁢ u a ≡ ( u 1 , … , u d ) = χ - 1 ⁢ ( x ) .$

In this case, the metric tensor $g a ⁢ b$ is given by $g α ⁢ β = e α ⋅ e β$ for $1 ≤ α , β ≤ d$ . The contravariant tangent basis is given by ${ e 1 , e 2 , … , e d }$ , where $e β = e α ⁢ g α ⁢ β = ( ∂ α ⁡ χ ) ⁢ g α ⁢ β$ (see [16]). Sometimes, we express $g a ⁢ b ≡ g = J T ⁢ J$ , where $J = [ e 1 , … , e d ]$ is an $n × d$ matrix.

Given a vector $v ∈ R n$ , it is in the tangent space $T x ⁢ ( Γ )$ if there exists a (contravariant) vector $v a$ such that $v ⁢ ( x ) = v α ⁢ e α ∘ χ - 1 ⁢ ( x )$ . Alternatively, one can write it in terms of a co-vector $v a$ and the contravariant basis, $v ⁢ ( x ) = v α ⁢ e α ∘ χ - 1 ⁢ ( x )$ . Moreover, any covariant (contravariant) vector $v a$ ( $v a$ ) has a corresponding extrinsic version given by $v = v α ⁢ e α$ ( $v = v α ⁢ e α$ ). We define the tangent bundle

$T ⁢ ( Γ ) = { ( x , v ) ∣ x ∈ Γ , v ⁢ ( x ) ∈ T x ⁢ ( Γ ) } ;$

thus, we say $v ∈ T ⁢ ( Γ )$ if $v ⁢ ( x ) ∈ T x ⁢ ( Γ )$ for every $x ∈ Γ$ ; in this case, we write $v : Γ → T ⁢ ( Γ )$ .

Next, we introduce extrinsic differential operators via their intrinsic counterpart, starting with the surface gradient $∇ Γ ⁡ f : Γ → T ⁢ ( Γ )$ defined in local coordinates by

(A.2)

$( ∇ Γ ⁡ f ) ∘ χ = ( ∇ α ⁡ f ) ⁢ g α ⁢ β ⁢ e β T = ∂ α ⁡ ( f ∘ χ ) ⁡ g α ⁢ β ⁢ ( ∂ β ⁡ χ ) T ≡ ∇ ⁡ ( f ∘ χ ) ⁡ g - 1 ⁢ J T .$

The (covariant) surface Hessian (a symmetric tensor) is given by

(A.3)

$( ∇ Γ ⁡ ∇ Γ ⁡ f ) ∘ χ := e μ ⁢ g μ ⁢ α ⁢ [ ∇ α ⁡ ∇ β ⁡ f ] ⁢ g β ⁢ ρ ⁢ e ρ T = e μ ⁢ g μ ⁢ α ⁢ [ ∂ α ⁡ ∂ β ⁡ ( f ∘ χ ) - ∂ γ ⁡ ( f ∘ χ ) ⁡ Γ α ⁢ β γ ] ⁢ g β ⁢ ρ ⁢ e ρ T .$

A.2.3 Special Case of a Surface

Suppose $d = 2$ and $n = 3$ . We have the following integration by parts relation:

$∫ Γ f ⁢ ∇ Γ ⋅ v ⁢ d S = ∫ ∂ ⁡ Γ f ⁢ v ⋅ n ⁢ d s - ∫ Γ ( ∇ Γ ⁡ f ) ⋅ v ⁢ d S ,$

$∫ Γ ( div Γ ⁢ r ) ⋅ ∇ Γ ⁡ f ⁢ d ⁢ S = ∫ ∂ ⁡ Γ ( n T ⁢ r ) ⋅ ∇ Γ ⁡ f ⁢ d ⁢ s - ∫ Γ r : ∇ Γ ⁡ ∇ Γ ⁡ f ⁢ d ⁢ S ,$

where we suppress the 𝑔 dependence in the differential measure and 𝒏 is the extrinsic conormal vector of

$∂ ⁡ Γ$ given by

$n ∘ χ | ∂ ⁡ U = n β ⁢ e β | n β ⁢ e β | ,$

where $| a |$ denotes the Euclidean length of the vector $a ∈ R n$ . Next, let 𝒕 be the unit tangent vector of a 1-𝑑 curve $Υ ⊂ Γ$ with conormal vector 𝒏, where $Υ = χ ⁢ ( Y )$ and $Y ⊂ U$ . In local coordinates, it is given by

$t ∘ χ | Y = t α ⁢ e α | t α ⁢ e α | ,$

where $t a$ is the (contravariant) tangent vector of 𝑌. Furthermore, let $ν : Γ → R 3$ be the surface unit normal vector of Γ, which satisfies $n = t × ν$ (see [43]) on $∂ ⁡ Γ$ . With the ambient space $R 3$ available, the tangent space projection $P : R 3 → R 3$ , defined on Γ, is given by

(A.4)

$P = I - ν ⊗ ν = t ⊗ t + n ⊗ n ,$

and note that (in local coordinates) $J ⁢ g - 1 ⁢ J T = P ∘ χ$ (see [43]).

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Received: 2020-08-12

Revised: 2021-06-03

Accepted: 2021-06-05

Published Online: 2021-06-23

Published in Print: 2022-01-01

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DOI: doi.org/10.1515/cmam-2020-0123

Keywords for this article

Surfaces With Boundary; Mesh-Dependent Norms; Non-Conforming Method; Surface Finite Elements