Abstract
We present the nonconforming virtual element method for the fourth-order singular perturbation problem. The virtual element proposed in this paper is a variant of the C0-continuous nonconforming virtual element presented in our previous work and allows to compute two different projection operators that are used for the construction of the discrete scheme. We show the optimal convergence in the energy norm for the nonconforming virtual element method. Further, the lowest order nonconforming method is proved to be uniformly convergent with respect to the perturbation parameter. Finally, we verify the convergence for the nonconforming virtual element method by some numerical tests.
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Acknowledgments
We would like to thank the anonymous reviewers, because their suggestions enrich our results.
Funding
This work is partially supported by National Natural Science Foundation of China (11701522, 11601124), Research Foundation for Advanced Talents of Henan University of Technology (2018BS013) and the scholarship from China Scholarship Council (201907045004).
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Communicated by: Long Chen
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Appendix
Appendix
First by using the bubble functions, we show some inverse inequalities for the local virtual space \(\widetilde {V_{h}^{K}}\) on every \(K\in \mathcal T_{h}\), of which the definition can be found in Section 3. Similar ideas can be found in [14, 15]. These inverse inequalities are used to prove the interpolation error estimates for the C0-continuous H2-nonconforming virtual element.
Lemma A.1
For every given\(K\in \mathcal T_{h}\), it holds that
Proof
For a given \(K\in \mathcal T_{h}\), let \(\lambda _{e}\in \mathbb P_{1}(K)\) be the function associated with the edge e of K defined by setting λe = −α(x −xe) ⋅nK/|e| such that λe = 0 on e, where the constant α > 0 is chosen to make sure \(\|\lambda _{e}\|_{\infty ,K}=1\). bK is the bubble function on K obtained by multiplying all the edge functions λe, so bK vanishes on ∂K and bK ≥ 0 in K where we have also used the fact that K is convex. With the help of bK, for any given \(v\in \widetilde {V_{h}^{K}}\), we have
Observing the fact that \({\Delta }^{2}v\in \mathbb P_{k-2}(K)\), we use the inverse inequality on polynomial space [7] to obtain
which leads to the inverse inequality (A.1). □
Further, we use the edge bubble function to prove the so-called trace inverse inequality for the local virtual space \(\widetilde {V_{h}^{K}}\) on every \(K\in \mathcal T_{h}\).
Lemma A.2
For every given\(K\in \mathcal T_{h}\)and e ⊂ ∂K, it holds that
Proof
For any given edge e of K, let be = bK/λe where bK is the bubble function on K and λe the edge function defined in the proof of Lemma A.1. Obviously it holds that be = 0 on ∂K∖e and \(b_{e}\in \mathbb P_{n-1}(K)\) in K, where n is the number of edges of K. For any given \(v\in \widetilde {V_{h}^{K}}\), the norm equivalence on polynomial space on edge yields
since \({\Delta } v\in \mathbb P_{k-2}(e)\) on e. Moreover, we observe that
Thus, we have beΔv ∈ H1(K). Further, the trace inequality and the Poincaré-Friedrichs inequality [7] imply
which, together with inequality (A.3), leads to
Let \(A=b_{e}{\Delta }\widetilde {V_{h}^{K}}\) and w = beΔv ∈ A. Because A is a finite dimensional subspace of L2(K), it holds
where the constant C is independent of hK. Generally, for a finite dimensional subspace VK ⊂ L2(K), it holds the norm equivalence
Using the generalized scaling argument, it is easy to show the constants c and C are indegendent of hK.
By using integration by parts, we have
For Δw, it holds the following identity:
Thus, we have
For all the terms on the right hand side of the above inequality, the inverse inequality on polynomial space and Lemma A.1 imply
Collecting up the above inequalities, we obtain
Therefore, observing w = beΔv, we obtain
which, together with inequality (A.4), yields (A.2). □
We introduce the global space for the H1-conforming virtual element [4] defined by
where
In [4], it has been shown that the following interpolation error estimates holds.
Lemma A.3
For every\(w\in {H_{0}^{1}}({\Omega })\cap H^{s}({\Omega })\)with 2 ≤ s ≤ k + 1, there exists a interpolation function\({w_{I}^{c}}\in U_{h}\)satisfying
Further, it holds
Next, we show an inverse inequality for the local virtual space \({U_{h}^{K}}\) on every \(K\in \mathcal T_{h}\) by using the bubble function defined in the proof of Lemma A.1.
Lemma A.4
For every given\(K\in \mathcal T_{h}\), it holds that
Proof
For any given \(K\in \mathcal T_{h}\) and \(v\in {U_{h}^{K}}\), let ϕ = ∇v, then |ϕ|1, K = |v|2, K and Δϕ = ∇(Δv). Further, we have
Here the constant C in the first inequality can be also shown to be independent of hK by the similar argument in the proof of Lemma A.2. Thus, the inverse inequality on polynomial space and the fact that \({\Delta } v\in \mathbb P_{k-2}(K)\) imply
which leads to the inverse inequality (A.5). □
With the above preparations, we show an approximation result for the global virtual space \(\widetilde V_{h}\) which is defined by the local space \(\widetilde {V_{h}^{K}}\) as follows:
To this end, we define the interpolation \(\tilde w_{I}\in \widetilde V_{h}\) for any \(w\in {H_{0}^{2}}({\Omega })\) by requiring that the values of the degrees of freedom (3.5)–(3.8) of \(\tilde w_{I}\) are equal to the corresponding ones of w. Then we have the following interpolation error estimates.
Lemma A.5
For every\(w\in {H_{0}^{2}}({\Omega })\cap H^{s}({\Omega })\)with 2 ≤ s ≤ k + 1, it holds that
Proof
From Green’s formula [19, 22] it holds that
where \(K\in \mathcal T_{h}\). For the details, see [33, Remark 4.2]. Thus, for any given \(K\in \mathcal T_{h}\), we have
Observing the fact that \({w_{I}^{c}}\) and \(\tilde w_{I}\) belongs to \(\mathbb P_{k}(e)\) on each edge e of K and are uniquely determined by the same degrees of freedom (3.5)–(3.6) of w, we have
Then we obtain
which, together with the interpolation properties of \(\tilde w_{I}\), implies
For the first term in Eq. (A.6), we use the inverse inequality (A.1) and Lemma A.3 to obtain
For the second term in Eq. (A.6), we use the trace inequality, Lemmas A.2–A.4 and Lemma 3.3 to obtain
Substituting (A.7)–(A.8) into (A.6), we obtain
which, together with the triangle inequality, inverse inequality (A.5), Lemma 3.3, and Lemma A.3, yields
By using the Poincaré-Friedrichs inequality [7] and the interpolation properties of \(\tilde w_{I}\), we further obtain
The proof is complete. □
Finally, we show the proof of Lemma 3.2 as follows.
The proof of Lemma 3.2 Observing the fact that the projection operator \({\Pi }_{\Delta }^{K}\) is uniquely determined by the degrees of freedom (3.11)–(3.14) of wI which are also the degrees of freedom of \(\tilde w_{I}\), we have
By similar arguments in the proof of Eq. (A.6), the interpolation properties of \(\tilde w_{I}\) and wI imply
For convenience, let \(\phi ={\Delta }^{2}(\tilde w_{I}-w_{I})\in \mathbb P_{k-2}(K)\). Recalling the definition (3.9) of \({W_{h}^{K}}\), the property of \({\Pi }_{\Delta }^{K}\), the Poincaré-Friedrichs inequality, and inverse inequality (A.1), we obtain
which, together with Lemma 3.3 and Lemma A.5, leads to
By using the triangle inequality and Lemma A.5, we obtain
By using the Poincaré-Friedrichs inequality [7] and the interpolation property of wI, we further obtain
The proof of Lemma 3.2 is complete.
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Zhang, B., Zhao, J. & Chen, S. The nonconforming virtual element method for fourth-order singular perturbation problem. Adv Comput Math 46, 19 (2020). https://doi.org/10.1007/s10444-020-09743-9
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DOI: https://doi.org/10.1007/s10444-020-09743-9