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A Spectral Method for Fourth-Order Mixed Inhomogeneous Boundary Value Problem in Three Dimensions

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Abstract

In this paper, we investigate spectral method for fourth- order mixed inhomogeneous boundary value problems in three dimensions. Some results on the three-dimensional Legendre approximation for fourth- order problem in Jacobi weighted Sobolev space are established, which improve and generalize the existing results, and play an important role in numerical solutions of partial differential equations. We also develop a lifting technique for fourth-order problems, with which we could handle mixed inhomogeneous boundary conditions easily. As examples of applications, spectral schemes are provided for three model problems with mixed inhomogeneous boundary conditions. The spectral accuracy in space of proposed algorithms are proved. Efficient implementations are presented. Numerical results demonstrate their high accuracy, and confirm the theoretical analysis well.

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Authors and Affiliations

  1. Henan University of Science and Technology, Luoyang, 471003, People’s Republic of China

    Tian-jun Wang

Authors
  1. Tian-jun Wang
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Corresponding author

Correspondence to Tian-jun Wang.

Additional information

This work is supported in part by NSF of China N.11371123, N.11571151 and N.11171227, Fund of Henan Education Commission N.14B110021 and The Doctor Fund of Henan University of Science and Technology N.09001263.

Appendix

Appendix

We shall change the inhomogeneous boundary value problem (3.1) to a boundary value problem with homogeneous Dirichlet boundary condition (3.6) on \(\partial \Omega \). For this purpose, we introduce some auxiliary functions. Firstly, let

$$\begin{aligned} f_-(x)= & {} \dfrac{x^3-3x+2}{4},\quad f_+(x)=\dfrac{-x^3+3x+2}{4},\\ h_-(x)= & {} \dfrac{x^3-x^2-x+1}{4},\quad h_+(x)=\dfrac{x^3+x^2-x-1}{4}. \end{aligned}$$

It can be checked that

$$\begin{aligned}&f_-(1)=\partial _x f_-(\pm 1)=0, \quad f_-(-1)=1,\quad f_+(-1)=\partial _x f_+(\pm 1)=0, \quad f_+(1)=1,\nonumber \\&h_-(\pm 1)=\partial _x h_-(1)=0, \quad \partial _x h_-(-1)=1,\quad h_+(\pm 1)=\partial _x h_+(-1)=0,\quad \partial _x h_+(1)=1.\nonumber \\ \end{aligned}$$
(5.1)

Secondly, the following three function \(W^0_F,W^0_E,W^0_V\) corresponds to the six faces, the twelve edges and the eight vertices, respectively.

We have from (3.2)-(3.5) and (5.1) that

$$\begin{aligned} {W_F^0}({{\varvec{x}}})= & {} f_+(x_1)g^0_1(x_2,x_3)+f_+(x_2)g^0_2(x_1,x_3) +f_+(x_3)g^0_3(x_1,x_2)\\&+\,f_-(x_1)g^0_4(x_2,x_3)+f_-(x_2)g^0_5(x_1,x_3) +f_-(x_3)g^0_6(x_1,x_2)\\ {W_E^0}({{\varvec{x}}})= & {} -f_+(x_1)f_+(x_2)g^0_{31}(x_3)-f_-(x_1)f_+(x_2)g^0_{32}(x_3)\\&-\,f_-(x_1)f_-(x_2)g^0_{33}(x_3)-f_+(x_1)f_-(x_2)g^0_{34}(x_3)\\&-\,f_+(x_1)f_+(x_3)g^0_{21}(x_2)-f_-(x_1)f_+(x_3)g^0_{22}(x_2)\\&-\,f_-(x_1)f_-(x_3)g^0_{23}(x_2)-f_+(x_1)f_-(x_3)g^0_{24}(x_2)\\&-\,f_+(x_2)f_+(x_3)g^0_{11}(x_1)-f_-(x_2)f_+(x_3)g^0_{12}(x_1)\\&-\,f_-(x_2)f_-(x_3)g^0_{13}(x_1)-f_+(x_2)f_-(x_3)g^0_{14}(x_1)\\ {W_V^0}({{\varvec{x}}})= & {} f_+(x_1)f_+(x_2)f_+(x_3)g^0_{31}(1)+f_-(x_1)f_+(x_2)f_+(x_3)g^0_{32}(1)\\&+\,f_-(x_1)f_-(x_2)f_+(x_3)g^0_{33}(1)+f_+(x_1)f_-(x_2)f_+(x_3)g^0_{34}(1)\\&+\,f_+(x_1)f_+(x_2)f_-(x_3)g^0_{31}(-1)+f_-(x_1)f_+(x_2)f_-(x_3)g^0_{32}(-1)\\&+\,f_-(x_1)f_-(x_2)f_-(x_3)g^0_{33}(-1)+f_+(x_1)f_-(x_2)f_-(x_3)g^0_{34}(-1). \end{aligned}$$

And the following five function \(W^1_F,W^1_{Ei},W^1_{Vi},i=1,2\) corresponds to the norm differential of the six faces, the twelve edges and the eight vertices of boundary , respectively,

$$\begin{aligned} {W_F^1}({{\varvec{x}}})= & {} +h_+(x_1)g^1_1(x_2,x_3)+h_+(x_2)g^1_2(x_1,x_3)+h_+(x_3)g^1_3(x_1,x_2)\\&\quad -\,h_-(x_1)g^1_4(x_2,x_3)-h_-(x_2)g^1_5(x_1,x_3)-h_-(x_3)g^1_6(x_1,x_2)\\ {W_{E1}^1}({{\varvec{x}}})= & {} -h_+(x_1)f_+(x_2)g^1_1(1,x_3)+h_-(x_1)f_+(x_2) g^1_4(1,x_3)\\&\quad -\,h_+(x_1)f_-(x_2) g^1_1(-1,x_3)+h_-(x_1)f_-(x_2)g^1_4(-1,x_3)\\&\quad -\,h_+(x_1)f_+(x_3)g^1_1(x_2,1)+h_-(x_1)f_+(x_3)g^1_4(x_2,1)\\&\quad -\,h_+(x_1)f_-(x_3)g^1_1(x_2,-1)+h_-(x_1)f_-(x_3)g^1_4(x_2,-1)\\&\quad -\,h_+(x_2)f_+(x_3)g^1_2(x_1,1)+h_-(x_2)f_+(x_3)g^1_5(x_1,1)\\&\quad -\,h_+(x_2)f_-(x_3)g^1_2(x_1,-1)+h_-(x_2)f_-(x_3)g^1_5(x_1,-1)\\&\quad -\,f_+(x_1)h_+(x_2)g^1_2(1,x_3)+f_+(x_1)h_-(x_2)g^1_5(1,x_3)\\&\quad -\,f_-(x_1)h_+(x_2)g^1_2(-1,x_3)+f_-(x_1)h_-(x_2)g^1_5(-1,x_3)\\&\quad -\,f_+(x_1)h_+(x_3)g^1_3(1,x_2)+f_+(x_1)h_-(x_3)g^1_6(1,x_2)\\&\quad -\,f_-(x_1)h_+(x_3)g^1_3(-1,x_2)+f_-(x_1)h_-(x_3)g^1_6(-1,x_2)\\&\quad -\,f_+(x_2)h_+(x_3)g^1_3(x_1,1)+f_+(x_2)h_-(x_3)g^1_6(x_1,1)\\&\quad -\,f_-(x_2)h_+(x_3)g^1_3(x_1,-1)+f_-(x_2)h_-(x_3)g^1_6(x_1,-1)\\ {W_{V1}^1}({{\varvec{x}}})= & {} +h_+(x_1)f_+(x_2)f_+(x_3)g^1_1(1.1)-h_-(x_1)f_+(x_2)f_+(x_3)g^1_4(1.1)\\&\quad +\,h_+(x_1)f_+(x_2)f_-(x_3)g^1_1(1,-1)-h_-(x_1)f_+(x_2)f_-(x_3)g^1_4(1,-1)\\&\quad +\,h_+(x_1)f_-(x_2)f_+(x_3)g^1_1(-1,1)-h_-(x_1)f_-(x_2)f_+(x_3) g^1_4(-1,1)\\&\quad +\,h_+(x_1)f_-(x_2)f_-(x_3)g^1_1(-1,-1)-h_-(x_1)f_-(x_2)f_-(x_3)g^1_4(-1,-1)\\&\quad +\,f_+(x_1)h_+(x_2)f_+(x_3)g^1_2(1.1)-f_+(x_1)h_-(x_2)f_+(x_3)g^1_5(1.1)\\&\quad +\,f_+(x_1)h_+(x_2)f_-(x_3)g^1_2(1,-1)-f_+(x_1)h_-(x_2)f_-(x_3)g^1_5(1,-1)\\&\quad +\,f_-(x_1)h_+(x_2)f_+(x_3)g^1_2(-1,1)-f_-(x_1)h_-(x_2)f_+(x_3)g^1_5(-1,1)\\&\quad +\,f_-(x_1)h_+(x_2)f_-(x_3)g^1_2(-1,-1)-f_-(x_1)h_-(x_2)f_-(x_3)g^1_5(-1,-1)\\&\quad +\,f_+(x_1)f_+(x_2)h_+(x_3)g^1_3(1.1)-f_+(x_1)f_+(x_2)h_-(x_3)g^1_6(1.1)\\&\quad +\,f_+(x_1)f_-(x_2)h_+(x_3)g^1_3(1,-1)-f_+(x_1)f_-(x_2)h_-(x_3)g^1_6(1,-1)\\&\quad +\,f_-(x_1)f_+(x_2)h_+(x_3)g^1_3(-1,1)-f_-(x_1)f_+(x_2)h_-(x_3) g^1_6(-1,1)\\&\quad +\,f_-(x_1)f_-(x_2)h_+(x_3)g^1_3(-1,-1)-f_-(x_1)f_-(x_2)h_-(x_3)g^1_6(-1,-1) \end{aligned}$$
$$\begin{aligned} {W_{E2}^1}({{\varvec{x}}})= & {} -\,h_+(x_1)h_+(x_2) \partial _{x_2}g^1_1(1,x_3)-h_+(x_1)h_-(x_2)\partial _{x_2}g^1_1(-1,x_3)\\&+\,h_-(x_1)h_+(x_2)\partial _{x_2}g^1_4(1,x_3)+h_-(x_1)h_-(x_2)\partial _{x_2}g^1_4(-1,x_3)\\&-\,h_+(x_1)h_+(x_3)\partial _{x_1}g^1_3(1,x_2)+h_+(x_1)h_-(x_3)\partial _{x_1}g^1_6(1,x_2)\\&-\,h_-(x_1)h_+(x_3)\partial _{x_1}g^1_3(-1,x_2)+h_-(x_1)h_-(x_3)\partial _{x_1}g^1_3(-1,x_2)\\&-\,h_+(x_2)h_+(x_3) \partial _{x_3}g^1_2(x_1,1)-h_+(x_2)h_-(x_3)\partial _{x_3}g^1_2(x_1,-1)\\&+\,h_-(x_2)h_+(x_3) \partial _{x_3}g^1_5(x_1,1)+h_-(x_2)h_-(x_3)\partial _{x_3}g^1_5(x_1,-1)\\ {W_{V2}^1}({{\varvec{x}}})= & {} +\,h_+(x_1)h_+(x_2)f_+(x_3)\partial _{x_2}g^1_1(1.1)+h_+(x_1)h_-(x_2)f_+(x_3)\partial _{x_2}g^1_1(-1,1)\\&-\,h_-(x_1)h_+(x_2)f_+(x_3)\partial _{x_2}g^1_4(1.1)-h_-(x_1)h_-(x_2)f_+(x_3)\partial _{x_2}g^1_4(-1,1)\\&+\,h_+(x_1)h_+(x_2)f_-(x_3)\partial _{x_2}g^1_1(1,-1)+h_+(x_1)h_-(x_2)f_-(x_3)\partial _{x_2}g^1_1(-1,-1)\\&-\,h_-(x_1)h_+(x_2)f_-(x_3)\partial _{x_2}g^1_4(1,-1)-h_-(x_1)h_-(x_2)f_-(x_3)\partial _{x_2}g^1_4(-1,-1)\\&+\,h_+(x_1)f_+(x_2)h_+(x_3)\partial _{x_1}g^1_3(1.1)-h_+(x_1)f_+(x_2)h_-(x_3)\partial _{x_1}g^1_6(1.1)\\&+\,h_-(x_1)f_+(x_2)h_+(x_3)\partial _{x_1}g^1_3(-1,1)-h_-(x_1)f_+(x_2)h_-(x_3)\partial _{x_1}g^1_6(-1,1)\\&+\,h_+(x_1)f_-(x_2)h_+(x_3)\partial _{x_1}g^1_3(1,-1)-h_+(x_1)f_-(x_2)h_-(x_3)\partial _{x_1}g^1_6(1,-1)\\&+\,h_-(x_1)f_-(x_2)h_+(x_3)\partial _{x_1}g^1_3(-1,-1)-h_-(x_1)f_-(x_2)h_-(x_3)\partial _{x_1}g^1_6(-1,-1)\\&+\,f_+(x_1)h_+(x_2)h_+(x_3)\partial _{x_3}g^1_2(1.1)+f_+(x_1)h_+(x_2)h_-(x_3)\partial _{x_3}g^1_2(1,-1)\\&-\,f_+(x_1)h_-(x_2)h_+(x_3)\partial _{x_3}g^1_5(1.1)-f_+(x_1)h_-(x_2)h_-(x_3)\partial _{x_3}g^1_5(1,-1)\\&+\,f_-(x_1)h_+(x_2)h_+(x_3)\partial _{x_3}g^1_2(-1,1)+f_-(x_1)h_+(x_2)h_-(x_3)\partial _{x_3}g^1_2(-1,-1)\\&-\,f_-(x_1)h_-(x_2)h_+(x_3)\partial _{x_3}g^1_5(-1,1)-f_-(x_1)h_-(x_2)h_-(x_3)\partial _{x_3}g^1_5(-1,-1)\\&+\,h_+(x_1)h_+(x_2)h_+(x_3)\partial _{x_1}\partial _{x_2}g^1_3(1.1)-h_+(x_1)h_+(x_2)h_-(x_3)\partial _{x_1}\partial _{x_2}g^1_6(1.1)\\&+\,h_+(x_1)h_-(x_2)h_+(x_3)\partial _{x_1}\partial _{x_2}g^1_3(1,-1)\!-\!h_+(x_1)h_-(x_2)h_-(x_3)\partial _{x_1}\partial _{x_2}g^1_6(1,-1)\\&+\,h_-(x_1)h_+(x_2)h_+(x_3)\partial _{x_1}\partial _{x_2}g^1_3(-1,1)\!-\!h_-(x_1)h_+(x_2)h_-(x_3)\partial _{x_1}\partial _{x_2}g^1_6(-1,1)\\&+\,h_-(x_1)h_-(x_2)h_+(x_3)\partial _{x_1}\partial _{x_2}g^1_3(-1,-1)\\&-h_-(x_1)h_-(x_2)h_-(x_3)\partial _{x_1}\partial _{x_2}g^1_6(-1,-1). \end{aligned}$$

Finally, set

$$\begin{aligned} W_B^0({{\varvec{x}}})= & {} W_F^0({{\varvec{x}}}) +W_E^0({{\varvec{x}}})+W_V^0({{\varvec{x}}}),\\ W^1_B({{\varvec{x}}})= & {} W^1_F({{\varvec{x}}}) +W^1_{E1}({{\varvec{x}}})+W^1_{V1}({{\varvec{x}}})+ W^1_{E2}({{\varvec{x}}})+W^1_{V2}({\varvec{x}}). \end{aligned}$$

Then, define the function corresponding to the boundary \(\partial \Omega \) by

$$\begin{aligned} W_B({{\varvec{x}}})=W_B^0({{\varvec{x}}})+W^1_B({{\varvec{x}}}). \end{aligned}$$

It can be checked that \({W}_B({{\varvec{x}}})=W({{\varvec{x}}})\) and \(\partial _{n} {W}_B({{\varvec{x}}})=\partial _{n} W({{\varvec{x}}})\) on \(\partial \Omega \).

Next, we construct the function \({\overline{W}}_B({\varvec{x}})\), with which shall change the inhomogeneous boundary value problem (3.31) to a boundary value problem with homogeneous Dirichlet boundary condition on \(\partial ^*\Omega \). Let \(W_B^0({{\varvec{x}}})\) be the same as before, and the following five function \(W^1_F,W^1_{Ei},W^1_{Vi},i=1,2\) corresponds to the three faces, the nine edges and the seven vertices concerning the norm differential of boundary \(\partial ^*\Omega \), respectively. Also, we have from (3.2)–(3.5) and (5.1) that

$$\begin{aligned} {W_F^1}({{\varvec{x}}})= & {} +\,h_+(x_1)g^1_1(x_2,x_3)+h_+(x_2)g^1_2 (x_1,x_3)+h_+(x_3)g^1_3(x_1,x_2),\\ {W_{E1}^1}({{\varvec{x}}})= & {} -\,h_+(x_1)f_+(x_2)g^1_1(1,x_3) -h_+(x_1)f_-(x_2)g^1_1(-1,x_3)\\&-\,h_+(x_1)f_+(x_3)g^1_1(x_2,1) -h_+(x_1)f_-(x_3)g^1_1(x_2,-1)\\&-\,h_+(x_2)f_+(x_3)g^1_2(x_1,1) -h_+(x_2)f_-(x_3)g^1_2(x_1,-1)\\&-\,f_+(x_1)h_+(x_2)g^1_2(1,x_3) -f_-(x_1)h_+(x_2)g^1_2(-1,x_3)\\&-\,f_+(x_1)h_+(x_3)g^1_3(1,x_2) -f_-(x_1)h_+(x_3)g^1_3(-1,x_2)\\&-\,f_+(x_2)h_+(x_3)g^1_3(x_1,1) -f_-(x_2)h_+(x_3)g^1_3(x_1,-1),\\ {W_{V1}^1}({{\varvec{x}}})= & {} +\,h_+(x_1)f_+(x_2)f_+(x_3)g^1_1(1.1) +h_+(x_1)f_+(x_2)f_-(x_3)g^1_1(1,-1)\\&+\,h_+(x_1)f_-(x_2)f_+(x_3) g^1_1(-1,1) +h_+(x_1)f_-(x_2)f_-(x_3)g^1_1(-1-,1)\\&+\,f_+(x_1)h_+(x_2)f_+(x_3)g^1_2(1.1) +f_+(x_1)h_+(x_2)f_-(x_3)g^1_2(1,-1)\\&+\,f_-(x_1)h_+(x_2)f_+(x_3)g^1_2(-1,1) +f_-(x_1)h_+(x_2)f_-(x_3)g^1_2(-1,-1)\\&+\,f_+(x_1)f_+(x_2)h_+(x_3)g^1_3(1.1) +f_+(x_1)f_-(x_2)h_+(x_3)g^1_3(1,-1)\\&+\,f_-(x_1)f_+(x_2)h_+(x_3)g^1_3(-1,1) +f_-(x_1)f_-(x_2)h_+(x_3)g^1_3(-1,-1),\\ {W_{E2}^1}({{\varvec{x}}})= & {} -\,h_+(x_1)h_+(x_2) \partial _{x_2}g^1_1(1,x_3) -h_+(x_1)h_+(x_3)\partial _{x_1}g^1_3(1,x_2)\\&-\,h_+(x_2)h_+(x_3) \partial _{x_3}g^1_2(x_1,1),\\ {W_{V2}^1}({{\varvec{x}}})= & {} +\,h_+(x_1)h_+(x_2)f_+(x_3) \partial _{x_2}g^1_1(1.1) +h_+(x_1)h_+(x_2)f_-(x_3) \partial _{x_2}g^1_1(1,-1)\\&+\,h_+(x_1)f_+(x_2)h_+(x_3)\partial _{x_1}g^1_3(1.1) +h_+(x_1)f_-(x_2)h_+(x_3)\partial _{x_1}g^1_3(1,-1)\\&+\,f_+(x_1)h_+(x_2)h_+(x_3)\partial _{x_3}g^1_2(1.1) +f_-(x_1)h_+(x_2)h_+(x_3)\partial _{x_3}g^1_2(1,-1)\\&+\,h_+(x_1)h_+(x_2)h_+(x_3)\partial _{x_1}\partial _{x_2}g^1_3(1.1). \end{aligned}$$

Set

$$\begin{aligned} W^1_B({{\varvec{x}}})=W^1_F({{\varvec{x}}}) +W^1_{E1}({\varvec{x}})+W^1_{V1}({{\varvec{x}}})+W^1_{E2} ({{\varvec{x}}})+W^1_{V2}({{\varvec{x}}}). \end{aligned}$$

Then, define the function corresponding to the boundary \(\partial \Omega \) and \(\partial ^*\Omega \) by

$$\begin{aligned} \overline{W}_B({{\varvec{x}}})=W_B^0({{\varvec{x}}})+W^1_B({{\varvec{x}}}). \end{aligned}$$

It can be checked that \({\overline{W}}_B({{\varvec{x}}})=W({{\varvec{x}}})\) on \(\partial \Omega \) and \(\partial _{n} {\overline{W}}_B({\varvec{x}})=\partial _{n} W({{\varvec{x}}})\) on \(\partial ^* \Omega \).

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Wang, Tj. A Spectral Method for Fourth-Order Mixed Inhomogeneous Boundary Value Problem in Three Dimensions. J Sci Comput 67, 1247–1271 (2016). https://doi.org/10.1007/s10915-015-0106-4

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