Hessian discretisation method for fourth-order semi-linear elliptic equations: applications to the von Kármán and Navier–Stokes models

Abstract

This paper deals with the Hessian discretisation method (HDM) for fourth-order semi-linear elliptic equations with a trilinear non-linearity. The HDM provides a generic framework for the convergence analysis of several numerical methods, such as the conforming and nonconforming finite element methods (ncFEMs) and methods based on gradient recovery (GR) operators. The Adini ncFEM and GR method, a specific scheme that is based on cheap, local reconstructions of higher-order derivatives from piecewise linear functions, are analysed for the first time for fourth-order semi-linear elliptic equations with trilinear non-linearity. Four properties, namely, the coercivity, consistency, limit-conformity and compactness, enable the convergence analysis in HDM framework that does not require any regularity of the exact solution. Two important problems in applications, namely, the Navier–Stokes equations in stream function vorticity formulation and the von Kármán equations of plate bending, are discussed. Results of numerical experiments are presented for the Morley and Adini ncFEMs, and the GR method.

Appendix. Proof of the properties associated with ncFEMs and GR method

This section deals with the proofs of the properties associated with the ncFEMs and GR methods stated in Theorems 4.5 and 4.6. For notational convenience, the first three properties are proved using the generic notation ${\mathcal {D}}$ (without index).

1.1 A.1 Nonconforming FEM

The preliminaries below are useful to prove the convergence of the Adini and Morley HDM for non-linear equations.

Let h_σ denote the diameter of σ and for all $w \in H^{1}({\mathcal M})$ (the broken Sobolev space H¹ on the mesh), define $\Vert {{\cdot }}\Vert _{dG,{\mathcal M}}$ by

$$ {\Vert w\Vert}_{dG,\mathcal{M}}^{2}:={\Vert \nabla_{\mathcal{M}}w \Vert^{2}}+ \underset{{\sigma} \in \mathcal{F}}{\sum}^{}\frac{1}{h_{{\sigma}}}{\Vert \llbracket w \rrbracket \Vert}_{L^{2}(\sigma)}^{2}. $$

Lemma A.1

[12, Theorems 5.3, 5.6] Let ℓ ≥ 0 be a non-negative integer. It holds

• (i) [Discrete Sobolev embedding] For all $v_{h} \in \mathbb {P}_{\ell }({\mathcal M})$, $\Vert {v_{h}}\Vert _{L^{4}} \lesssim \Vert {v_{h}}\Vert _{dG,{\mathcal M}}.$

• (ii) [Discrete Rellich theorem] Let $({\mathcal M}_{m})_{m\in \mathbb {N}}$ be sequence of regular triangular or rectangular meshes, whose diameter h_m tend to 0 as $m\to \infty$. For all $m\in \mathbb {N}$, let $v_{m}\in \mathbb {P}_{\ell }({{\mathscr{M}}}_{m})$. If $(\Vert {v_{m}}\Vert _{dG,{{\mathscr{M}}}_{m}})_{m\in \mathbb {N}}$ is bounded, then, for all 1 ≤ q < 2^∗ (where 2^∗ is a Sobolev exponent of 2), the sequence $(v_{m})_{m\in \mathbb {N}}$ is relatively compact in L^q(Ω).

• (iii) [27, Lemma 6.2] Let $w\in H^{1}({\mathcal M})$. If for all ${\sigma } \in {\mathcal {F}}$ there exists x_σ ∈ σ such that ⟦w⟧(x_σ) = 0, then $\Vert {w}\Vert _{dG,{\mathcal M}} \lesssim \Vert {\nabla _{{\mathcal M}}w}\Vert .$

Lemma A.2

[14, 27] Let ${\mathcal D}$ be a HD for the Morley (resp. Adini) ncFEM in the sense of Definition 3.2 (resp. Definition 3.3). Then,

$$ \begin{array}{@{}rcl@{}} &\forall v_{{\mathcal{D}}} \in X_{{\mathcal D},0}, \Vert{{\Pi}_{{\mathcal D}} v_{{\mathcal D}}}\Vert \lesssim \Vert{{\mathcal{H}_{\mathcal{D}}} v_{{\mathcal D}}}\Vert ,\quad \forall \xi \in H^{2}({\Omega};{\mathbb{R}}^{2 \times 2}), W_{{\mathcal D}}(\xi) \lesssim h \Vert{\xi}\Vert_{H^{2}({\Omega};{\mathbb{R}}^{2 \times 2})}. \end{array} $$

We now prove the four properties associated with the Morley and Adini ncFEMs.

Proof of Theorem 4.5

(i) The Morley element.

• (i) Let $v_{{\mathcal D}} \in X_{{\mathcal D},0}$. Since $\llbracket \nabla _{{\mathcal D}} v_{{\mathcal D}} \rrbracket =0$ at the midpoints of the edges, Lemma A.1(i) and Lemma A.1(iii) lead to

  $$ \Vert{\nabla_{\mathcal D} v_{\mathcal D}}\Vert_{L^4} \lesssim \Vert{\nabla_{\mathcal D} v_{\mathcal D}}\Vert_{dG,{\mathcal M}} \lesssim \Vert{{\mathcal{H}}_{{\mathcal D}}v_{\mathcal D}}\Vert. $$

  (A.1)

  The combination of this and Lemma A.2(i) prove $C_{{\mathcal D}} \lesssim 1$.

• (ii) For $\varphi \in H^{3}({\Omega })\cap {H^{2}_{0}}({\Omega })$, the standard interpolant [10, 21] satisfies

  $$ \begin{array}{@{}rcl@{}} &\Vert{I_{h}\varphi-\varphi}\Vert\lesssim h^{3} \Vert{\varphi }\Vert_{H^{3}({\Omega})}, \Vert{\nabla_{{\mathcal{M}}} I_{h}\varphi-\nabla \varphi}\Vert_{L^{4}} \lesssim h^{\frac{3}{2}} \Vert{\varphi}\Vert_{H^{3}({\Omega})}, \\ &\text{ and }\Vert{{\mathcal{H}}_{{\mathcal M}} I_{h}\varphi-{\mathcal{H}}\varphi}\Vert \lesssim h \Vert{\varphi }\Vert_{H^{3}({\Omega})}. \end{array} $$

  (A.2)

  Hence, selecting $w \in X_{{\mathcal D},0}$ corresponding to the degrees of freedom of I_hφ in the definition of $S_{{\mathcal D}}(\varphi )$ yields the result on consistency.

• (iii) For all ${K} \in {\mathcal M}$ and for any ${\sigma } \in {{{\mathcal {F}}}_K}$, denote the unit vector normal to σ outward K by n_{K, σ}. An integration by parts shows

  $$ \begin{array}{@{}rcl@{}} \left| {\int}_{{\Omega}} \left( \nabla_{{\mathcal D}} v_{{\mathcal D}} \cdot \phi + {\Pi}_{{\mathcal D}} v_{{\mathcal D}} \text{div} \phi \right)\text{d}\boldsymbol{x} \right|& =\left|\underset{{K} \in {\mathcal M}}{\sum}\underset{{\sigma} \in {{{\mathcal{F}}}_K}}{\sum} {\int}_{{\sigma}}(\phi \cdot n_{{K},{\sigma}}){\Pi}_{{\mathcal D}} v_{{\mathcal D}} \text{d} s(\boldsymbol{x})\right|. \end{array} $$

  (A.3)

  The proof of the result on $\widehat {W}_{{\mathcal D}}$ then follows from a slightly modified version of [21, Lemma 3.5]. However, for the sake of completeness, we provide a proof. Let V₁ be the space of all globally continuous piecewise linear functions and let ${\Pi }_{1}:V_{h} \rightarrow V_{1}$ be the interpolation operator such that π₁v_h equal to v_h at the vertices of all triangle K, v_h ∈ V_h. Then

  $$ \underset{{K} \in {\mathcal M}}{\sum}\underset{{\sigma} \in {{{\mathcal{F}}}_{K}}}{\sum} {\int}_{{\sigma}}^{}(\phi \cdot n_{{K},{\sigma}}){\Pi}_{\mathcal D} v_{\mathcal D} \text{d} s(\boldsymbol{x})=\underset{{K} \in {\mathcal M}}{\sum}\underset{{\sigma} \in {{{\mathcal{F}}}_{K}}}{\sum} {\int}_{{\sigma}}P(v_h-{\Pi}_1{v_h})\text{d} s(\boldsymbol{x}), $$

  (A.4)

  where P = ϕ ⋅ n_{K, σ} and $v_{h}={\Pi }_{{\mathcal D}} v_{{\mathcal D}}$. Note that a change of variables yields

  $$ {\int}_{{\sigma}\in {{{\mathcal{F}}}_{K}}}P(v_h-{\Pi}_1{v_h})\text{d} s(\boldsymbol{x})=|{\sigma}|{\int}_{\hat{{\sigma}} \in {\mathcal F}_{\hat{{K}}}}\widehat{P}(\widehat{v_h}-\widehat{{\Pi}_1}{\widehat{v_h}})\text{d} s(\boldsymbol{x}), $$

  (A.5)

  where ${\widehat {{K}}}$ is the reference finite element. The continuous trace inequality [12], the discrete trace inequality [12, Lemma 1.46], an interpolation estimate [10] and Young’s inequality show

  $$ \left|{\int}_{\widehat{{\sigma}} \in {\mathcal F}_{\widehat{{K}}}}\widehat{P}(\widehat{v_{h}}-{\Pi}_{1}{\widehat{v_{h}}})\text{d} s(\boldsymbol{x})\right|\lesssim \left( |{\widehat{P}}|_{0,\hat{K}}+|{\widehat{P}}|_{1,\widehat{K}}\right)|{\widehat{v}}|_{2,\widehat{K}}, $$

  where $|\cdot |_{\ell ,\widehat {K}}$ denotes the seminorm on $H^{\ell }(\widehat {K})$, ℓ ≥ 0. Substitute the above estimate in (A.5), transform from $\widehat {K}$ to K using $|\hat {\psi }|_{\ell ,\hat {K}} \lesssim h^{\ell -1}|\psi |_{\ell ,K}$ for ψ ∈ H^ℓ(K) ([10, Theorem 3.1.2]), sum over all the edges and then use (A.4) to obtain

  $$ \begin{array}{@{}rcl@{}} \left|\underset{{K} \in {\mathcal M}}{\sum}\underset{{\sigma} \in {{{\mathcal{F}}}_{K}}}{\sum} {\int}_{{\sigma}}^{}(\phi \cdot n_{{K},{\sigma}}){\Pi}_{{\mathcal D}} v_{{\mathcal D}} \text{d} s(\boldsymbol{x})\right|\lesssim (h \Vert{\phi}\Vert+h^{2} \Vert{\nabla\phi}\Vert )\Vert{{\mathcal{H}_\mathcal{D}} v_{{\mathcal D}}}\Vert. \end{array} $$

  This in (A.3) reads

  $$ \left| {\int}_{{\Omega}} \left( \nabla_{{\mathcal D}} v_{{\mathcal D}} \cdot \phi + {\Pi}_{{\mathcal D}} v_{{\mathcal D}} \text{div} \phi \right)\text{d}\boldsymbol{x} \right|\lesssim (h \Vert{\phi}\Vert+h^{2} \Vert{\nabla\phi}\Vert)\Vert{{\mathcal{H}_\mathcal{D}} v_{{\mathcal{D}}}}\Vert. $$

  The above estimate with Lemma A.2 leads to the required estimate on $\widehat {W}_{{\mathcal {D}}}$ and $W_{{\mathcal {D}}}$ and thus establish limit-conformity.

• (iv) Let a sequence $u_{m} \in X_{{\mathcal D}_{m},0}$ be such that $(\Vert {u_{m}}\Vert _{{\mathcal D}_{m}})_{m \in \mathbb {N}}$ is bounded. Since $\llbracket {\Pi }_{{\mathcal D}_{m}} u_{m} \rrbracket$ = 0 at the edge vertices, Lemma A.1(iii) and estimate (A.1) yield $\Vert {{\Pi }_{{\mathcal {D}}_m} u_{m}}\Vert _{dG,{\mathcal M}_{m}} \lesssim \Vert {\nabla _{{\mathcal {D}}_m} u_{m}}\Vert \lesssim \Vert {\nabla _{{\mathcal {D}}_m} u_{m}}\Vert _{L^{4}}\lesssim \Vert {{{\mathscr{H}}}_{{\mathcal {D}}_m}u_{m}}\Vert .$ At the edge midpoints $\llbracket \nabla _{{\mathcal {D}}_m} u_{m} \rrbracket =0$ and consequently, Lemma A.1(iii) with $w=\nabla _{{\mathcal {D}}_m} u_{m}$ shows $\Vert {\nabla _{{\mathcal {D}}_m} u_{m}}\Vert _{dG,{\mathcal M}_{m}}$ $\lesssim \Vert {{{\mathscr{H}}}_{{\mathcal {D}}_m}u_{m}}\Vert .$ Use the fact that $(\Vert {u_{m}}\Vert _{{\mathcal D}_{m}})_{m \in \mathbb {N}}$ is bounded to deduce $({\Pi }_{{\mathcal D}_{m}}u_{m})_{m \in \mathbb {N}}$ and $(\nabla _{{\mathcal D}_{m}}u_{m})_{m \in \mathbb {N}}$ are bounded in the $\Vert {\cdot }\Vert _{dG,{\mathcal M}_{m}}$ norm. Lemma A.1(ii) then gives the relative compactness of $({\Pi }_{{\mathcal D}_{m}}u_{m})_{m \in \mathbb {N}}$ in L²(Ω), and of $(\nabla _{{\mathcal D}_{m}}u_{m})_{m \in \mathbb {N}}$ in $L^{4}({\Omega };{\mathbb {R}}^{d})$.

(ii) The Adini element.

• (i) Since $\nabla _{{\mathcal D}} v_{{\mathcal D}}$ is continuous at the vertices of elements in ${\mathcal M}$ and $\nabla _{{\mathcal D}} v_{{\mathcal D}}$ vanish at vertices along ∂Ω, $\llbracket \nabla _{{\mathcal D}} v_{{\mathcal D}} \rrbracket =0$ at the vertices. As a consequence, Lemma A.1 (i)–(iii) leads to $\Vert {\nabla _{{\mathcal D}} v_{{\mathcal D}}}\Vert _{L^{4}} \lesssim \Vert {\nabla _{{\mathcal D}} v_{{\mathcal D}}}\Vert _{dG,{\mathcal M}} \lesssim \Vert {{{\mathscr{H}}}_{{\mathcal D}}v_{{\mathcal D}}}\Vert .$ This and Lemma A.2 concludes that $C_{{\mathcal D}} {\lesssim 1}$.

• (ii) The standard interpolant satisfies (A.2) and hence yields the desired estimate on the consistency measure $S_{{\mathcal D}}$.

• (iii) Any ${\sigma } \in {\mathcal F}$ is associated to a fixed orientation of the unit normal vector n_σ on σ. For $K \in {\mathcal M}$ and ${\sigma }\in {\mathcal F}_{K}$, set n_σ := n_{K, σ}. Apply integration by parts in each cell to obtain

  $$ \left| {\int}_{{\Omega}} \left( \nabla_{{\mathcal D}} v_{{\mathcal D}} \cdot \phi + {\Pi}_{{\mathcal D}} v_{{\mathcal D}} \text{div} \phi \right)\text{d}\boldsymbol{x} \right|=\left|\underset{{\sigma} \in {\mathcal F}}{\sum} {\int}_{{\sigma}}^{}(\phi \cdot n_{\sigma})\llbracket {\Pi}_{{\mathcal{D}}} v_{{\mathcal{D}}} \rrbracket \text{d} s(\boldsymbol{x})\right|. $$

  Here, the jump has been chosen in a compatible way with the orientation determined by n_σ. Since ${\Pi }_{{\mathcal {D}}} v_{{\mathcal {D}}} \in {H^{1}_{0}}({\Omega })\cap C(\overline {{\Omega }})$, $\llbracket {\Pi }_{{\mathcal D}} v_{{\mathcal D}} \rrbracket =0$, which implies $\widehat {W}_{{\mathcal D}}(\phi )=0$. This and Lemma A.2 yields an estimate on the limit-conformity measures, namely $\widehat {W}_{{\mathcal D}}$ and $W_{{\mathcal D}}$.

• (iv) The compactness proof follows as for the Morley element using the fact that $\llbracket {\Pi }_{{\mathcal D}} v_{{\mathcal D}} \rrbracket =0$ and $\llbracket \nabla _{{\mathcal D}} v_{{\mathcal D}} \rrbracket =0$ at the vertices.

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