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

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

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,

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:

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

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

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

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

A.2.3 Special Case of a Surface

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

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

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

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