Error analysis of approximate operators for a particle method based on Voronoi diagram

Hajime Koba; Kazuki Sato

doi:10.1515/anly-2022-1104

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Error analysis of approximate operators for a particle method based on Voronoi diagram

Hajime Koba and Kazuki Sato

Published/Copyright: February 28, 2023

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From the journal Analysis Volume 43 Issue 4

Abstract

This paper considers several approximate operators used in a particle method based on a Voronoi diagram. We introduce and study our approximate operators on gradient and Laplace operators. We derive error estimates for these approximate operators by applying our weight functions. The key idea of deriving our error estimates is to divide the integration region into a ring-shaped area and some areas. In the appendix, we give an exemplary application of the main results of this paper.

Keywords: Error analysis; error estimates; particle method; Voronoi diagram; Voronoi decomposition

MSC 2010: 33F05

A Appendix: Applications of main results

We state an application of the main results of this paper. We consider the following case:

w ⁢ ( x ) = { 1 , x ∈ B ¯ h , δ , 0 , x ∈ ℝ 2 ∖ B ¯ h , δ .

It is easy to check that L w = 0 and that for each 0 < q < p ,

∥ w ( a k - ⋅ ) ∥ L 1 ⁢ ( B p ⁢ ( a k ) ∖ B q ⁢ ( a k ) ) = π ( p 2 - q 2 ) ,

∥ w ( a k - ⋅ ) / | a k - ⋅ | ∥ L 1 ⁢ ( B p ⁢ ( a k ) ∖ B q ⁢ ( a k ) ) = 2 π ( p - q ) ,

∥ | a k - ⋅ | w ( a k - ⋅ ) ∥ L 1 ⁢ ( B p ⁢ ( a k ) ∖ B q ⁢ ( a k ) ) = 2 ⁢ π 3 ( p 3 - q 3 ) ,

∥ | a k - ⋅ | 2 w ( a k - ⋅ ) ∥ L 1 ⁢ ( B p ⁢ ( a k ) ∖ B q ⁢ ( a k ) ) = π 2 ( p 4 - q 4 ) .

In this section, we assume that δ = r σ / 2 and

∥ w ( x - ⋅ ) ∥ L 1 ⁢ ( B h ⁢ ( a k ) ∖ σ k ) = ∥ w ( x - ⋅ ) ∥ L 1 ⁢ ( B h , δ ⁢ ( a k ) ) ,

Suppose that there is C * > 1 such that

h = C * ⁢ r σ .

Assume that ℛ ⁢ ( a k , λ ⁢ h ) ≠ ∅ for some 0 < λ < 1 . Using Theorems 1.3–1.6 and

∥ ⋅ ∥ L 1 ⁢ ( σ k ∖ B δ ⁢ ( a k ) ) ≤ ∥ ⋅ ∥ L 1 ⁢ ( B r σ ⁢ ( a k ) ∖ B δ ⁢ ( a k ) ) ,

we have the following corollary.

Corollary B.

For each f ∈ C 3 ⁢ ( Ω ¯ H ) ,

| f ⁢ ( a k ) - Π ~ h ⁢ f ⁢ ( a k ) | ≤ ( C * ⁢ r σ + r σ ) ⁢ | f | C 1 + 3 / 2 C * 2 - 1 / 4 ⁢ | f | C 0 ,

(B.1) | ∇ ⁡ f ⁢ ( a k ) - ∇ ~ h ⁢ f ⁢ ( a k ) | ≤ 4 ⁢ C * ⁢ r σ ⁢ | f | C 2 + ( 8 λ ⁢ C * + 8 ⁢ ( λ ⁢ C * + 1 ) 2 + 1 C * 2 - 1 / 4 ) ⁢ | f | C 1 ,

| Δ ⁢ f ⁢ ( a k ) - Δ ~ h ⁢ f ⁢ ( a k ) | ≤ 24 ⁢ C * ⁢ r σ ⁢ | f | C 3 + 1 3 ⁢ r σ ⁢ ( C * 4 - 1 / 16 )

× ( 8 + 24 ⁢ ( λ ⁢ C * + 1 ) 2 + 56 ⁢ C * 3 - 7 3 ⁢ ( C * 4 - 1 / 16 ) + C * ⁢ ( 64 ⁢ C * 3 - 8 ) C * 2 + 1 / 4 + 16 ⁢ C * 3 - 2 λ ⁢ C * ) ⁢ | f | C 1 ,

| Δ ⁢ f ⁢ ( a k ) - □ ~ h ⁢ f ⁢ ( a k ) | ≤ 24 ⁢ C * ⁢ r σ ⁢ | f | C 3 + ( 32 ⁢ λ ⁢ C * + 24 r σ ⁢ ( C * 2 - 1 / 4 ) + 32 λ ⁢ C * ⁢ r σ ⁢ ( C * + 1 / 2 ) ) ⁢ | f | C 1 .

Moreover, the following two assertions hold:

If r σ = 10 - 5 ⁢ m , C * = 10 4 ⁢ m , λ = 10 - 2 ⁢ m for some m ∈ ℕ , then

| f ⁢ ( a k ) - Π ~ h ⁢ f ⁢ ( a k ) | ≤ ( 1 10 m + 1 10 5 ⁢ m ) ⁢ | f | C 1 + 1 10 8 ⁢ m - 1 ⁢ | f | C 0 ,

(B.2) | ∇ ⁡ f ⁢ ( a k ) - ∇ ~ h ⁢ f ⁢ ( a k ) | ≤ 4 10 m ⁢ | f | C 2 + 1 10 2 ⁢ m - 1 ⁢ | f | C 1 ,

| Δ ⁢ f ⁢ ( a k ) - Δ ~ h ⁢ f ⁢ ( a k ) | ≤ 24 10 m ⁢ | f | C 3 + 1 10 m - 1 ⁢ | f | C 1 ,

| Δ ⁢ f ⁢ ( a k ) - □ ~ h ⁢ f ⁢ ( a k ) | ≤ 24 10 m ⁢ | f | C 3 + 1 10 m - 2 ⁢ | f | C 1 .
If r σ = 10 - 2 , C * = 4 , λ = 1 2 , then

| f ⁢ ( a k ) - Π ~ h ⁢ f ⁢ ( a k ) | ≤ 1 20 ⁢ | f | C 1 + 1 10 ⁢ | f | C 0 ,

| ∇ ⁡ f ⁢ ( a k ) - ∇ ~ h ⁢ f ⁢ ( a k ) | ≤ 4 25 ⁢ | f | C 2 + 10 ⁢ | f | C 1 ,

| Δ ⁢ f ⁢ ( a k ) - Δ ~ h ⁢ f ⁢ ( a k ) | ≤ 24 25 ⁢ | f | C 3 + 300 ⁢ | f | C 1 ,

| Δ ⁢ f ⁢ ( a k ) - □ ~ h ⁢ f ⁢ ( a k ) | ≤ 24 25 ⁢ | f | C 3 + 1000 ⁢ | f | C 1 .

Proof of Corollary B.

We only show (B.1) and (B.2). Fix f ∈ C 3 ⁢ ( Ω ¯ H ) .

We first show (B.1). Since L w = 0 , it follows from Theorem 1.4 to see that

(B.3) | ∇ ⁡ f ⁢ ( a k ) - ∇ ~ h ⁢ f ⁢ ( a k ) | ≤ 4 ⁢ h ⁢ | f | C 2 + { 8 ⁢ r σ λ ⁢ h + 4 ⁢ ∥ w ( a k - ⋅ ) ∥ L 1 ⁢ ( σ k ∖ B δ ⁢ ( a k ) ) ∥ w ( a k - ⋅ ) ∥ L 1 ⁢ ( B h ⁢ ( a k ) ∖ B δ ⁢ ( a k ) ) + 8 ⁢ ∥ w ( a k - ⋅ ) ∥ L 1 ⁢ ( B λ ⁢ h + r σ ⁢ ( a k ) ∖ σ k ) ∥ w ( a k - ⋅ ) ∥ L 1 ⁢ ( B h ⁢ ( a k ) ∖ σ k ) } ⁢ | f | C 1 .

By the assumptions of Section A, we check that the right-hand side of (B.3) is greater than or equal to

4 ⁢ C * ⁢ r σ ⁢ | f | C 2 + ( 8 ⁢ r σ λ ⁢ C * ⁢ r σ + 4 ⁢ π ⁢ ( r σ 2 - r σ 2 / 4 ) π ⁢ ( C * 2 ⁢ r σ 2 - r σ 2 / 4 ) + 8 ⁢ π ⁢ { ( λ ⁢ C * ⁢ r σ + r σ ) 2 - r σ 2 / 4 } π ⁢ ( C * 2 ⁢ r σ 2 - r σ 2 / 4 ) ) ⁢ | f | C 1

= 4 ⁢ C * ⁢ r σ ⁢ | f | C 2 + ( 8 λ ⁢ C * + 8 ⁢ ( λ ⁢ C * + 1 ) 2 + 1 C * 2 - 1 / 4 ) ⁢ | f | C 1 .

Therefore, we obtain (B.1).

Next we show (B.2). A direct calculation shows that

4 ⁢ C * ⁢ r σ = 4 ⋅ 10 4 ⁢ m ⋅ 10 - 5 ⁢ m = 4 10 m

and that

8 λ ⁢ C * + 8 ⁢ ( λ ⁢ C * + 1 ) 2 + 1 C * 2 - 1 / 4 = 8 10 2 ⁢ m + 8 ⁢ ( 10 2 ⁢ m + 1 ) 2 + 1 10 8 ⁢ m - 1 / 4

≤ 8 10 2 ⁢ m + 1 10 2 ⁢ m ≤ 1 10 2 ⁢ m - 1 .

From (B.1), we have (B.2). Therefore, Corollary B is proved. ∎

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments to improve this paper.

References

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Received: 2022-09-14

Revised: 2023-01-19

Accepted: 2023-01-24

Published Online: 2023-02-28

Published in Print: 2023-11-01

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Articles in the same Issue

https://doi.org/10.1515/anly-2022-1104

Keywords for this article

Error analysis; error estimates; particle method; Voronoi diagram; Voronoi decomposition