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A New Family of Mixed Method for the Biharmonic Eigenvalue Problem Based on the First Order Equations of Hellan–Herrmann–Johnson Type

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Abstract

In this paper, we consider the numerical approximation of a biharmonic eigenvalue problem by introducing a new family of the mixed method. This method is based on a formulation where the fourth-order eigenproblem is recast as a system of four first-order equations. The optimal convergence rates with \(2k+2\) (\(k\ge 0\) is the degree of the polynomials) of eigenvalue approximation are theoretically derived and numerically verified. The optimal or sub-optimal convergences of the other unknowns are theoretically proved. The new numerical schemes based on the deduced problems can be of lower complicacy, and the framework is fit for various fourth-order eigenvalue problems.

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Acknowledgements

The authors would like to thank both referees for their valuable comments and helpful suggestions which improved this paper.

Funding

The work of Y. Li was partially supported by the Scientific Research Plan of Tianjin Municipal Education Committee (2017KJ236), the Humanity and Social Science Youth Foundation of Ministry of Education of China (19YJC630199), and the National Natural Science Foundations of China (NSFC 12101447, 12271395). The work of M. Xie was partially supported by the National Natural Science Foundations of China (NSFC 12001402, 12071343, 12271400). The work of C. Xiong was partially supported by the Beijing Municipal Natural Science Foundation (BMNSF4182059) and the National Natural Science Foundations of China (NSFC 12271035).

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Authors and Affiliations

  1. Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin, 300222, China

    Yu Li

  2. Center for Applied Mathematics, Tianjin University, Tianjin, 300072, China

    Manting Xie

  3. Department of Mathematics, Beijing Key Laboratory on Mathematical Characterization, Analysis, and Applications of Complex Information, Beijing Institute of Technology, Beijing, 102488, China

    Chunguang Xiong

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  1. Yu Li
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Correspondence to Chunguang Xiong.

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Appendix: The proof of Lemma 3.1

Appendix: The proof of Lemma 3.1

In order to get the best possible estimates, we assume the following elliptic regularity result:

$$\begin{aligned} \Vert u^f\Vert _4\le C\Vert f\Vert _0 \end{aligned}$$
(A.1)

Then we have error equations as follows

$$\begin{aligned} (\varvec{q}^f-\varvec{q}_h^f,\varvec{p}_h) + (u^f-u_h^f,\nabla \cdot \varvec{p}_h)&= 0,, \end{aligned}$$
(A.2a)
$$\begin{aligned} (\underline{\varvec{z}}^f-\underline{\varvec{z}}^f_h,\underline{\varvec{s}}_h) + (\varvec{q}^f-\varvec{q}^f_h,\nabla \cdot \underline{\varvec{s}}_h)&= 0,, \end{aligned}$$
(A.2b)
$$\begin{aligned} -(\varvec{w}^f-\varvec{w}^f_h,\varvec{m}_h) + (\nabla \cdot (\underline{\varvec{z}}^f- \underline{\varvec{z}}^f_h),\varvec{m}_h)&= 0, \end{aligned}$$
(A.2c)
$$\begin{aligned} (\nabla \cdot (\varvec{w}^f-\varvec{w}^f_h), v_h)&= 0, \end{aligned}$$
(A.2d)

for all \((v_h, \varvec{p}_h,\underline{\varvec{s}}_h,\varvec{m}_h) \in V_h\times \varvec{Q}_h\times \underline{\varvec{Z}}_h \times \varvec{W}_h \).

Now we begin to prove Lemma 3.1.

Proof

Using the dual equations (A.2) of the source problem (2.1), we have

$$\begin{aligned} (\Pi ^{V}_hu-u_h,\chi )= & {} (\Pi ^{V}_hu-u_h,\nabla \cdot \tilde{\varvec{w}})\ \ \qquad {(\text {by dual equation }\mathrm{(3.8d)})}\\= & {} (\Pi ^V_hu-u_h,\nabla \cdot \varvec{\Pi }^{RT}\tilde{\varvec{w}})\ \ \quad {(\text {by } \mathrm{(3.1a)} )}\\= & {} -(\varvec{q}^f-\varvec{q}^f_h,\varvec{\Pi }^{RT}\tilde{\varvec{w}})\ \ \qquad (\text {by error equation} (\mathrm{A.2a}))\\= & {} -(\varvec{q}^f-\varvec{q}^f_h,\varvec{\Pi }^{RT}\tilde{\varvec{w}}-\tilde{\varvec{w}})-(\varvec{q}^f-\varvec{q}^f_h,\tilde{\varvec{w}}-\varvec{\Pi }^Q_h\tilde{\varvec{w}})-(\varvec{q}^f-\varvec{q}^f_h,\varvec{\Pi }^Q_h\tilde{\varvec{w}}). \end{aligned}$$

We express the last term \((\varvec{q}^f-\varvec{q}^f_h,\varvec{\Pi }^Q_h\tilde{\varvec{w}})\). By the dual equation (3.8c), we have

$$\begin{aligned} (\varvec{q}^f-\varvec{q}^f_h, \varvec{\Pi }^Q_h\tilde{\varvec{w}})= & {} (\varvec{q}^f-\varvec{q}^f_h, \varvec{\Pi }^Q_h\nabla \cdot {\tilde{\underline{\varvec{z}}}}) \quad (\text { by dual equation } (\mathrm{3.8c}))\\= & {} -(\varvec{q}^f-\varvec{q}^f_h,\nabla \cdot {\underline{\varvec{\Pi }}}^{RT}{\tilde{\underline{\varvec{z}}}})\ \ \quad (\text {by the commutative property }\varvec{\Pi }^Q_h)\\= & {} (\underline{\varvec{z}}^f-\underline{\varvec{z}}^f_h,{\underline{\varvec{\Pi }}}^{RT}{\tilde{\underline{\varvec{z}}}}-{\tilde{\underline{\varvec{z}}}}) + (\underline{\varvec{z}}^f-\underline{\varvec{z}}^f_h,{\tilde{\underline{\varvec{z}}}}).\ \ \quad (\text {by error equation } (\mathrm{A.2b})) \end{aligned}$$

Furthermore, using the integration by parts, we have

$$\begin{aligned} (\underline{\varvec{z}}^f-\underline{\varvec{z}}^f_h,{\tilde{\underline{\varvec{z}}}})= & {} (\underline{\varvec{z}}^f-\underline{\varvec{z}}^f_h,\nabla \tilde{\varvec{q}}) = (\nabla \cdot (\underline{\varvec{z}}^f-\underline{\varvec{z}}^f_h),\tilde{\varvec{q}})\ \ \quad (\text {by }\tilde{\varvec{q}}|_{\partial \Omega }=0)\\= & {} (\nabla \cdot (\underline{\varvec{z}}^f-\underline{\varvec{z}}^f_h),\tilde{\varvec{q}}-\varvec{\Pi }^{Q}_h\tilde{\varvec{q}}) + (\nabla \cdot (\underline{\varvec{z}}^f-\underline{\varvec{z}}^f_h),\varvec{\Pi }^{Q}_h\tilde{\varvec{q}}) \\= & {} ({\varvec{w}}^f,\tilde{\varvec{q}}-\varvec{\Pi }^{Q}_h\tilde{\varvec{q}}) + ({\varvec{w}}^f-{\varvec{w}}^f_h,\varvec{\Pi }^{Q}_h\tilde{\varvec{q}}) \ \ \quad (\text {by error equation } (\mathrm{A.2b}) \text { and } (\mathrm{2.1c}))\\= & {} ({\varvec{w}}^f-\varvec{\Pi }^{RT}{\varvec{w}}^f,\tilde{\varvec{q}}-\varvec{\Pi }^{Q}_h\tilde{\varvec{q}}) + ({\varvec{w}}^f-\varvec{\Pi }^{RT}{\varvec{w}}^f, \varvec{\Pi }^{Q}_h\tilde{\varvec{q}}) + (\varvec{\Pi }^{RT}{\varvec{w}}^f-{\varvec{w}}^f_h,\varvec{\Pi }^Q_h\tilde{\varvec{q}}). \end{aligned}$$

Next, we express the second term and the third term

$$\begin{aligned} ({\varvec{w}}^f-\varvec{\Pi }^Q_h{\varvec{w}}^f,\varvec{\Pi }^{RT}\tilde{\varvec{q}})= & {} ({\varvec{w}}^f-\varvec{\Pi }^{RT}{\varvec{w}}^f, \varvec{\Pi }^Q_h\tilde{\varvec{q}}^f - \tilde{\varvec{q}}) +({\varvec{w}}^f-\varvec{\Pi }^{RT}{\varvec{w}}^f,\nabla {{\tilde{u}}}) \\= & {} ({\varvec{w}}^f-\varvec{\Pi }^{RT}{\varvec{w}}^f, \varvec{\Pi }^Q_h\tilde{\varvec{q}} - \tilde{\varvec{q}}) +(\nabla \cdot {\varvec{w}}^f-\nabla \cdot \varvec{\Pi }^{RT}{\varvec{w}}^f,{{\tilde{u}}}) \\= & {} ({\varvec{w}}^f-\varvec{\Pi }^{RT}{\varvec{w}}^f, \varvec{\Pi }^Q_h\tilde{\varvec{q}} - \tilde{\varvec{q}}) +(f-\Pi ^{V}_h\nabla \cdot {\varvec{w}}^f,{{\tilde{u}}}) \\= & {} ({\varvec{w}}^f-\varvec{\Pi }^{RT}{\varvec{w}}^f, \varvec{\Pi }^Q_h\tilde{\varvec{q}} - \tilde{\varvec{q}}) +(f-\Pi ^V_hf,{{\tilde{u}}}-\Pi ^V_h{{\tilde{u}}}). \end{aligned}$$

The last term vanishes. In fact, it follows from \({\varvec{w}}^f_h-\varvec{\Pi }^{RT}{\varvec{w}}^f\in \varvec{Q}_h\cap \varvec{W}_h\) and \(\nabla \cdot ({\underline{\varvec{\Pi }}}^{RT}{\varvec{w}}^f-{\varvec{w}}^f_h) = 0\) that

$$\begin{aligned} (\varvec{\Pi }^{RT}{\varvec{w}}^f-{\varvec{w}}^f_h, \varvec{\Pi }^{Q}_h\tilde{\varvec{q}})= & {} (\varvec{\Pi }^{RT}{\varvec{w}}^f-{\varvec{w}}^f_h,\tilde{\varvec{q}}^f) = (\varvec{\Pi }^{RT}{\varvec{w}}^f-{\varvec{w}}^f_h,\nabla {{\tilde{u}}}) \\= & {} (\nabla \cdot (\varvec{\Pi }^{RT}{\varvec{w}}^f-{\varvec{w}}^f_h),{{\tilde{u}}}) = 0 \end{aligned}$$

by the integration by parts, \({{\tilde{u}}}|_{\partial \Omega }=0\) and (A.2d). Combining the all above steps implies the desired result. \(\square \)

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Li, Y., Xie, M. & Xiong, C. A New Family of Mixed Method for the Biharmonic Eigenvalue Problem Based on the First Order Equations of Hellan–Herrmann–Johnson Type. J Sci Comput 93, 66 (2022). https://doi.org/10.1007/s10915-022-02024-1

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