Stabilised Finite Element Methods for Ill-Posed Problems with Conditional Stability

Abstract

In this paper we discuss the adjoint stabilised finite element method introduced in Burman (SIAM J Sci Comput 35(6):A2752–A2780, 2013) and how it may be used for the computation of solutions to problems for which the standard stability theory given by the Lax-Milgram Lemma or the Babuska-Brezzi Theorem fails. We pay particular attention to ill-posed problems that have some conditional stability property and prove (conditional) error estimates in an abstract framework. As a model problem we consider the elliptic Cauchy problem and provide a complete numerical analysis for this case. Some numerical examples are given to illustrate the theory.

Appendix

We will here give a proof that the inf-sup stability (22) holds also for the stabilisation (18). We do not track the depedence on γ _D and γ _S .

Proposition 3

Let A _h [(⋅,⋅),(⋅,⋅)] be defined by  (13) with a _h (⋅,⋅), s _W (⋅,⋅) and s _V (⋅,⋅) defined by Eq.  (14) – (16) and  (18) (or  (19) for s _W (⋅,⋅)). Then the inf-sup condition  (22) is satisfied for the semi-norm  (30) .

Proof

We must prove that the L ²-stabilisation of the jump of the Laplacian gives sufficient control for the inf-sup stability of $\mathcal{L}u_{h}$ evaluated elementwise. It is well known [14] that for the quasi-interpolation operator defined in each node x _i by

$$\displaystyle{(I_{os}\varDelta u_{h})(x_{i}):= N_{i}^{-1}\sum _{ \{K:x_{i}\in K\}}\varDelta u_{h}(x_{i})\vert _{K},}$$

N _i : = card{K: x _i  ∈ K} the following discrete interpolation result holds

$$\displaystyle{ \|h(\varDelta u_{h} - I_{os}\varDelta u_{h})\|_{h}\leqslant C_{os}s_{V }^{S}(u_{ h},u_{h})^{\frac{1} {2} } }$$

(58)

as well as following the stabilities obtained using trace inequalities, inverse inequalities and the L ²-stability of I _os ,

$$\displaystyle{ \|h^{\frac{3} {2} }I_{os}\varDelta u_{h}\|_{\mathcal{F}} +\| h^{\frac{5} {2} }\partial _{n}I_{os}\varDelta u_{h}\|_{\mathcal{F}} +\| hI_{os}\varDelta u_{h}\|_{h} + \vert h^{2}I_{os}\varDelta u_{h}\vert _{s_{ X}}\lesssim \|h\varDelta u_{h}\|_{h}. }$$

(59)

First observe that by taking (v _h , w _h ) = (u _h , z _h ) we have

$$\displaystyle{\vert u_{h}\vert _{s_{V }}^{2} + \vert z_{ h}\vert _{s_{W}}^{2} = A_{ h}[(u_{h},z_{h}),(u_{h},z_{h})].}$$

Now let $w_{h}^{\mathcal{L}} = h^{2}I_{os}\mathcal{L}u_{h} = h^{2}(I_{os}\varDelta u_{h} + cu_{h})$, $v_{h}^{\mathcal{L}} = h^{2}I_{os}\mathcal{L}^{{\ast}}z_{h}$. Using (59) it is straightforward to show that

$$\displaystyle{ \|h^{\frac{3} {2} }I_{os}\mathcal{L}u_{h}\|_{\mathcal{F}} +\| h^{\frac{5} {2} }\partial _{n}I_{os}\mathcal{L}u_{h}\|_{\mathcal{F}} +\| hI_{os}\mathcal{L}u_{h}\|_{h} + \vert h^{2}I_{os}\mathcal{L}u_{h}\vert _{s_{ X}}\leqslant \tilde{C}_{os}\|h\mathcal{L}u_{h}\|_{h}. }$$

(60)

Now observe that (for a suitably chosen orientation of the normal on interior faces)

$$\displaystyle\begin{array}{rcl} a_{h}(u_{h},w_{h}^{\mathcal{L}})& =& \|h\mathcal{L}u_{ h}\|_{h}^{2} + (\mathcal{L}u_{ h},h^{2}(I_{ os}\mathcal{L}u_{h} -\mathcal{L}u_{h}))_{h} + \left < [\![\partial _{n}u_{h}]\!],h^{2}I_{ os}\mathcal{L}u_{h}\right >_{\mathcal{F}_{I}} {}\\ & & \quad + \left < \partial _{n}u_{h},h^{2}I_{ os}\mathcal{L}u_{h}\right >_{\varGamma _{N}} + \left < \partial _{n}h^{2}I_{ os}\mathcal{L}u_{h},u_{h}\right >_{\varGamma _{D}} {}\\ & \geqslant & \frac{1} {2}\|h\mathcal{L}u_{h}\|_{h}^{2} - 2\|h^{2}(I_{ os}\mathcal{L}u_{h} -\mathcal{L}u_{h})\|_{h}^{2} - 2\tilde{C}_{ os}^{-2}s_{ V }^{D}(u_{ h},u_{h}) {}\\ & & \geqslant \frac{1} {2}\|h\mathcal{L}u_{h}\|_{h}^{2} - 2C_{ os}^{2}s_{ V }^{S}(u_{ h},u_{h}) - 2\tilde{C}_{os}^{-2}s_{ V }^{D}(u_{ h},u_{h}) {}\\ & & \phantom{\geqslant \frac{1} {2}\|h\mathcal{L}u_{h}\|_{h}^{2} - 2C_{ os}^{2}s_{ V }^{S}(u_{ h},}\geqslant \frac{1} {2}\|h\mathcal{L}u_{h}\|_{h}^{2} - 2(C_{ os}^{2} +\tilde{ C}_{ os}^{-2})\vert u_{ h}\vert _{s_{V }}^{2} {}\\ \end{array}$$

and

$$\displaystyle{s_{W}(z_{h},w_{h}^{\mathcal{L}})\geqslant -\tilde{ C}_{ os}^{-2}\vert z_{ h}\vert _{s_{W}}^{2} -\frac{1} {4}\|h\mathcal{L}u_{h}\|_{h}^{2}.}$$

Similarly

$$\displaystyle{a_{h}(v_{h}^{\mathcal{L}},z_{ h})\geqslant \frac{1} {2}\|h\mathcal{L}^{{\ast}}z_{ h}\|_{h}^{2} - 2(C_{ os}^{2} +\tilde{ C}_{ os}^{-2})\vert z_{ h}\vert _{s_{W}}^{2}}$$

and

$$\displaystyle{s_{V }(u_{h},v_{h}^{\mathcal{L}})\geqslant -\tilde{ C}_{ os}^{-2}\vert u_{ h}\vert _{s_{V }}^{2} -\frac{1} {4}\|h\mathcal{L}^{{\ast}}z_{ h}\|_{h}^{2}.}$$

It follows that for some c ₁, c ₂ > 0 there holds

$$\displaystyle{\vert (u_{h},z_{h})\vert _{\mathcal{L}}^{2}\lesssim A_{ h}[(u_{h},z_{h}),(u_{h} + c_{1}w_{h}^{\mathcal{L}},z_{ h} + c_{2}v_{h}^{\mathcal{L}})].}$$

We conclude by observing that by inverse inequalities and (60) we have the stability

$$\displaystyle{\vert (u_{h} + c_{1}w_{h}^{\mathcal{L}},z_{ h} + c_{2}v_{h}^{\mathcal{L}})\vert _{ \mathcal{L}}\lesssim \vert (u_{h},z_{h})\vert _{\mathcal{L}}.}$$

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