Numerical Approximation of Gaussian Random Fields on Closed Surfaces

A Proof of Lemma 4.4

We follow the argument from [18, Section 4]. Note that from the definition of 𝒬 k - s , we have

where ℰ l : 𝕍 ~ ⁢ ( 𝒯 ) → 𝕍 ⁢ ( 𝒯 ) is given by

with μ l := e y l and y l are the sinc quadrature points; see (4.20). The decomposition of the operator ℰ l

is instrumental in the error analysis below.

The next result recall estimate for some terms in the above decomposition. We refer to [18, Lemma 4.5] for more details.

We also note that the arguments leading to (3.18) but using expansions based on the discrete eigenpairs imply that if r ∈ ( n - 1 2 , 2 ⁢ s ) and μ > 0 we have

for any F ~ ∈ 𝕍 ~ ⁢ ( 𝒯 ) .

To prove Lemma 4.4, it suffices to show that

which we do now by estimating each term separately.

We start with S 2 and let r ∈ ( n - 1 2 , 2 ⁢ s ) . Thanks to (A.4), the geometric error (4.5), and (A.5), we obtain

The estimate of ∥ L ~ 𝒯 - r 2 ⁢ W ~ Ψ ∥ L 2 ⁢ ( Ω ; L 2 ⁢ ( γ ) ) provided by Lemma 4.2 in conjunction with - s + r 2 < 0 , yield

which is the desired estimate in disguised (with C ⁢ ( h ) ≲ 1 ).

For S 1 , we first estimate the discrepancy between U ~ 𝒯 := T ~ 𝒯 ⁢ F ~ and U 𝒯 := T 𝒯 ⁢ σ ⁢ P ⁢ F ~ for any F ~ ∈ 𝕍 ~ ⁢ ( 𝒯 ) . By definition, see (4.3), U 𝒯 satisfies

for any V ∈ 𝕍 ⁢ ( 𝒯 ) . In turn, the change of variable formula (4.4) together with the definition of U ~ 𝒯 imply

upon realizing that P - 1 ⁢ V ∈ 𝕍 ~ ⁢ ( 𝒯 ) . Consequently, we find that

where the error matrix 𝐄 ~ satisfies ∥ 𝐄 ~ ∥ L ∞ ⁢ ( γ ) ≲ h 2 (see, e.g., [17, Corollary 33]). We now set V = P ⁢ U ~ 𝒯 - U 𝒯 so that with Cauchy-Schwarz inequality and the geometric consistency (4.5), we get

We next estimate ∥ ℰ l 1 ⁢ W ~ Ψ ∥ L 2 ⁢ ( Ω ; L 2 ⁢ ( Γ ) ) distinguishing two cases.

Case 1. If μ l > 1 , we apply successively (A.3), (A.7) and (A.2) (with p = 1 and q = - min ⁡ { 1 , r } ) to get

where we used that μ l - 1 2 ≤ μ l r 2 - 1 if r > 1 .

Case 2. If 0 < μ l ≤ 1 , we set again q = - min ⁡ { 1 , r } . Using to the relation L 𝒯 ⁢ ( μ l ⁢ I + L 𝒯 ) - 1 = I - μ l ⁢ ( μ l ⁢ I + L 𝒯 ) - 1 and (A.4), we have

Moreover, (A.7) implies that

while (A.2) with p = - q reads

Gathering the above three estimates yields

and as a consequence

Recall that q = - min ⁡ ( 1 , r ) and so we now argue depending on whether r ≤ 1 (which can only happen when n = 2 ) and r > 1 . When r ≤ 1 , we have q = - r and thus S 1 ≲ h 2 by Lemma 4.2. For r > 1 , i.e. q = - 1 , we write r = 1 + ϵ with ϵ ∈ ( 0 , 2 ⁢ s - 1 ) and we invoke an inverse inequality to write

where we applied Lemma 4.2 for the second inequality. To conclude, we choose ϵ = 2 ⁢ s - 1 ln ⁡ ( h - 1 ) yielding S 1 ≲ h 2 for n = 2 and S 1 ≲ ln ⁡ ( h - 1 ) ⁢ h 2 for n = 3 . The proof is now complete.