𝑯(curl
               2)-Conforming Spectral Element Method for Quad-Curl Problems

Lixiu Wang; Huiyuan Li; Zhimin Zhang

doi:10.1515/cmam-2020-0152

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𝑯(curl ²)-Conforming Spectral Element Method for Quad-Curl Problems

Lixiu Wang , Huiyuan Li und Zhimin Zhang

Veröffentlicht/Copyright: 6. Mai 2021

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Aus der Zeitschrift Computational Methods in Applied Mathematics Band 21 Heft 3

Abstract

In this paper, we propose an H ⁢ ( curl 2 ) -conforming spectral elements to solve the quad-curl problem on cubic meshes in three dimensions. Starting with generalized vectorial Jacobi polynomials, we first construct the basis functions of the H ⁢ ( curl 2 ) -conforming spectral elements using the contravariant transform together with the affine mapping from the reference cube onto each physical element. Falling into four categories, interior modes, face modes, edge modes, and vertex modes, these H ⁢ ( curl 2 ) -conforming basis functions are constructed in an arbitrarily high degree with a hierarchical structure. Next, H ⁢ ( curl 2 ) -conforming spectral element approximation schemes are established to solve the boundary value problem as well as the eigenvalue problem of quad-curl equations. Numerical experiments demonstrate the effectiveness and efficiency of the ℎ-version and the 𝑝-version of our H ⁢ ( curl 2 ) -conforming spectral element method.

Keywords: Cubic Spectral Element Method; Quad-Curl Problems

MSC 2010: 65N30; 35Q60; 65N15; 35B45

Funding source: China Postdoctoral Science Foundation

Award Identifier / Grant number: 2020M670117

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11871145

Award Identifier / Grant number: 11971016

Award Identifier / Grant number: 11871092

Award Identifier / Grant number: U1930402

Funding statement: The research of the first author is supported in part by China Postdoctoral Science Foundation (No. 2020M670117). The research of the second author is supported in part by the National Natural Science Foundation of China (NSFC 11871145 and NSFC 11971016). The research of the third author is supported in part by the National Natural Science Foundation of China (NSFC 11871092, NSFC 11926356, and NSAF U1930402).

A Trace Properties on the Vectorial Jacobi Basis

Lemma A.1

Let

ϕ | F = [ ϕ | F 1 , ϕ | F 2 , ϕ | F 3 , ϕ | F 4 , ϕ | F 5 , ϕ | F 6 ]

be the trace on six faces. It then holds that, for k ∈ N 0 ,

tr T ⁡ φ i , j , k | F = { [ 0 , 0 , 0 , 0 , 0 , 0 ] , i , j ∈ N 0 ∖ { 0 , 2 } , [ φ ^ - 1 , j , k ⁢ e z ^ h z ^ , 0 , 0 , 0 , 0 , 0 ] , i = 0 , j ∈ N 0 ∖ { 0 , 2 } , [ 0 , φ ^ - 1 , j , k ⁢ e z ^ h z ^ , 0 , 0 , 0 , 0 ] , i = 2 , j ∈ N 0 ∖ { 0 , 2 } , [ 0 , 0 , φ ^ i , - 1 , k ⁢ e z ^ h z ^ , 0 , 0 , 0 ] , j = 0 , i ∈ N 0 ∖ { 0 , 2 } , [ 0 , 0 , 0 , φ ^ i , - 1 , k ⁢ e z ^ h z ^ , 0 , 0 ] , j = 2 , i ∈ N 0 ∖ { 0 , 2 } , [ φ ^ - 1 , 0 , k ⁢ e z ^ h z ^ , 0 , φ ^ 0 , - 1 , k ⁢ e z ^ h z ^ , 0 , 0 , 0 ] , i = 0 , j = 0 , [ φ ^ - 1 , 2 , k ⁢ e z ^ h z ^ , 0 , 0 , φ ^ 0 , - 1 , k ⁢ e z ^ h z ^ , 0 , 0 ] , i = 0 , j = 2 , [ 0 , φ ^ - 1 , 0 , k ⁢ e z ^ h z ^ , φ ^ 2 , - 1 , k ⁢ e z ^ h z ^ , 0 , 0 , 0 ] , i = 2 , j = 0 , [ 0 , φ ^ - 1 , 2 , k ⁢ e z ^ h z ^ , 0 , φ ^ 2 , - 1 , k ⁢ e z ^ h z ^ , 0 , 0 ] , i = 2 , j = 2 .

It also holds for k ≥ 2 that

(A.1) tr T ⁡ ( ∇ × φ i , j , k ) | F = { [ 0 , 0 , 0 , 0 , 0 , 0 ] , i , j ∈ N 0 ∖ { 1 , 3 } , [ 0 , 0 , φ ^ i , - 1 , k ⁢ e x ^ h y ^ ⁢ h z ^ , 0 , 0 , 0 ] , i ∈ N 0 ∖ { 1 , 3 } , j = 1 , [ 0 , 0 , 0 , φ ^ i , - 1 , k ⁢ e x ^ h y ^ ⁢ h z ^ , 0 , 0 ] , i ∈ N 0 ∖ { 1 , 3 } , j = 3 , [ - φ ^ - 1 , j , k ⁢ e y ^ h x ^ ⁢ h z ^ , 0 , 0 , 0 , 0 , 0 ] , i = 1 , j ∈ N 0 ∖ { 1 , 3 } , [ 0 , - φ ^ - 1 , j , k ⁢ e y ^ h x ^ ⁢ h z ^ , 0 , 0 , 0 , 0 ] , i = 3 , j ∈ N 0 ∖ { 1 , 3 } , [ - φ ^ - 1 , 1 , k ⁢ e y ^ h x ^ ⁢ h z ^ , 0 , φ ^ 1 , - 1 , k ⁢ e x ^ h y ^ ⁢ h z ^ , 0 , 0 , 0 ] , i = 1 , j = 1 , [ - φ ^ - 1 , 3 , k ⁢ e y ^ h x ^ ⁢ h z ^ , 0 , 0 , φ ^ 1 , - 1 , k ⁢ e x ^ h y ^ ⁢ h z ^ , 0 , 0 ] , i = 1 , j = 3 , [ 0 , - φ ^ - 1 , 1 , k ⁢ e y ^ h x ^ ⁢ h z ^ , φ ^ 3 , - 1 , k ⁢ e x ^ h y ^ ⁢ h z ^ , 0 , 0 , 0 ] , i = 3 , j = 1 , [ 0 , - φ ^ - 1 , 3 , k ⁢ e y ^ h x ^ ⁢ h z ^ , 0 , φ ^ 3 , - 1 , k ⁢ e x ^ h y ^ ⁢ h z ^ , 0 , 0 ] , i = 3 , j = 3 ,

for k = 1 that

tr T ⁡ ( ∇ × φ i , j , 1 ) | F = { [ 0 , 0 , 0 , 0 , 0 , a i , j ] , i , j ∈ N 0 ∖ { 1 , 3 } , [ 0 , 0 , φ ^ i , - 1 , 1 ⁢ e x ^ h y ^ ⁢ h z ^ , 0 , 0 , a i , 1 ] , i ∈ N 0 ∖ { 1 , 3 } , j = 1 , [ 0 , 0 , 0 , φ ^ i , - 1 , 1 ⁢ e x ^ h y ^ ⁢ h z ^ , 0 , a i , 3 ] , i ∈ N 0 ∖ { 1 , 3 } , j = 3 , [ - φ ^ - 1 , j , 1 ⁢ e y ^ h x ^ ⁢ h z ^ , 0 , 0 , 0 , 0 , a 1 , j ] , i = 1 , j ∈ N 0 ∖ { 1 , 3 } , [ 0 , - φ ^ - 1 , j , 1 ⁢ e y ^ h x ^ ⁢ h z ^ , 0 , 0 , 0 , a 3 , j ] , i = 3 , j ∈ N 0 ∖ { 1 , 3 } , [ - φ ^ - 1 , 1 , 1 ⁢ e y ^ h x ^ ⁢ h z ^ , 0 , φ ^ 1 , - 1 , 1 ⁢ e x ^ h y ^ ⁢ h z ^ , 0 , 0 , a 1 , 1 ] , i = 1 , j = 1 , [ - φ ^ - 1 , 3 , 1 ⁢ e y ^ h x ^ ⁢ h z ^ , 0 , 0 , φ ^ 1 , - 1 , 1 ⁢ e x ^ h y ^ ⁢ h z ^ , 0 , a 1 , 3 ] , i = 1 , j = 3 , [ 0 , - φ ^ - 1 , 1 , 1 ⁢ e y ^ h x ^ ⁢ h z ^ , φ ^ 3 , - 1 , 1 ⁢ e x ^ h y ^ ⁢ h z ^ , 0 , 0 , a 3 , 1 ] , i = 3 , j = 1 , [ 0 , - φ ^ - 1 , 3 , 1 ⁢ e y ^ h x ^ ⁢ h z ^ , 0 , φ ^ 3 , - 1 , 1 ⁢ e x ^ h y ^ ⁢ h z ^ , 0 , a 3 , 3 ] , i = 3 , j = 3 ,

and for k = 0 that

(A.2) tr T ⁡ ( ∇ × φ i , j , 0 ) | F = { [ 0 , 0 , 0 , 0 , a i , j , 0 ] , i , j ∈ N 0 ∖ { 1 , 3 } , [ 0 , 0 , φ ^ i , - 1 , 0 ⁢ e x ^ h y ^ ⁢ h z ^ , 0 , a i , 1 , 0 ] , i ∈ N 0 ∖ { 1 , 3 } , j = 1 , [ 0 , 0 , 0 , φ ^ i , - 1 , 0 ⁢ e x ^ h y ^ ⁢ h z ^ , a i , 3 , 0 ] , i ∈ N 0 ∖ { 1 , 3 } , j = 3 , [ - φ ^ - 1 , j , 0 ⁢ e y ^ h x ^ ⁢ h z ^ , 0 , 0 , 0 , a 1 , j , 0 ] , i = 1 , j ∈ N 0 ∖ { 1 , 3 } , [ 0 , - φ ^ - 1 , j , 0 ⁢ e y ^ h x ^ ⁢ h z ^ , 0 , 0 , a 3 , j , 0 ] , i = 3 , j ∈ N 0 ∖ { 1 , 3 } , [ - φ ^ - 1 , 1 , 0 ⁢ e y ^ h x ^ ⁢ h z ^ , 0 , φ ^ 1 , - 1 , 0 ⁢ e x ^ h y ^ ⁢ h z ^ , 0 , a 1 , 1 , 0 ] , i = 1 , j = 1 , [ - φ ^ - 1 , 3 , 0 ⁢ e y ^ h x ^ ⁢ h z ^ , 0 , 0 , φ ^ 1 , - 1 , 0 ⁢ e x ^ h y ^ ⁢ h z ^ , a 1 , 3 , 0 ] , i = 1 , j = 3 , [ 0 , - φ ^ - 1 , 1 , 0 ⁢ e y ^ h x ^ ⁢ h z ^ , φ ^ 3 , - 1 , 0 ⁢ e x ^ h y ^ ⁢ h z ^ , 0 , a 3 , 1 , 0 ] , i = 3 , j = 1 , [ 0 , - φ ^ - 1 , 3 , 0 ⁢ e y ^ h x ^ ⁢ h z ^ , 0 , φ ^ 3 , - 1 , 0 ⁢ e x ^ h y ^ ⁢ h z ^ , a 3 , 3 , 0 ] , i = 3 , j = 3 ,

where

a i , j = J i - 2 , - 2 ( x ^ ) J j - 2 , - 2 ( y ^ ) ′ e x ^ h y ^ ⁢ h z ^ - J i - 2 , - 2 ( x ^ ) ′ J j - 2 , - 2 ( y ^ ) e y ^ h x ^ ⁢ h z ^ .

Proof

Note that

∇ ^ × φ ^ i , j , k ( x ^ , y ^ , z ^ ) = J i - 2 , - 2 ( x ^ ) J j - 2 , - 2 ( y ^ ) ′ J k - 1 , - 1 ( z ^ ) e x ^ - J i - 2 , - 2 ( x ^ ) ′ J j - 2 , - 2 ( y ^ ) J k - 1 , - 1 ( z ^ ) e y ^ .

Then the results can be obtained by the properties of the generalized Jacobi polynomials immediately, and we omit the details of the proof. ∎

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Received: 2020-09-21

Revised: 2021-01-25

Accepted: 2021-04-08

Published Online: 2021-05-06

Published in Print: 2021-07-01

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Cubic Spectral Element Method; Quad-Curl Problems

𝑯(curl 2)-Conforming Spectral Element Method for Quad-Curl Problems

Artikel

Abstract

A Trace Properties on the Vectorial Jacobi Basis

Proof

References

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𝑯(curl ²)-Conforming Spectral Element Method for Quad-Curl Problems