Abstract
We provide a posteriori error estimates in the energy norm for temporal semi-discretisations of wave maps into spheres that are based on the angular momentum formulation. Our analysis is based on novel weak–strong stability estimates which we combine with suitable reconstructions of the numerical solution. We present time-adaptive numerical simulations based on the a posteriori error estimators for solutions involving blow-up.
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Funding
Open Access funding enabled and organized by Projekt DEAL. J.G. thanks the German Research Foundation (DFG) for the support of the project via DFG grant GI 1131/1-1. E. M.-B. is funded by the DFG via the grant MA 7559/1-1 and appreciates the support.
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David Jakob Stonner and Elena Mäder-Baumdicker contributed equally to this work.
Appendix
Appendix
This appendix is devoted to the proof of Lemma 2. Here, each quantity that is defined continuously in time is to be understood as its restriction to \((t_n,t_{n+1})\). Let us begin by giving explicit formulae for certain parts of \(r_u\) and \(r_\omega \). The first such expressions make explicit the difference between some piecewise quadratic, globally continuous function and its piecewise linear interpolations:
1.1 Controlling \(|\widehat{u}|\), \(|u^*|\)
Let us now study how far away \(\widehat{u}, u^*\) are from maps into the sphere: If \(u^{n+1}-u^n\) is sufficiently small, a geometric argument implies
and, therefore,
Thus, the conditions in Lemma 2 imply
1.2 Estimating \(r_{u,1}\)
Since \(\tilde{u}(t_n)=\widehat{u}(t_n)\) and \(\tilde{w}(t_n)=\widehat{\omega }(t_n) \) we may rewrite \(r_{u,1} \) as
such that
This proves (44).
We obtain (45) by applying the product rule to (56) since \(|\widehat{u}(t,x)|\le 1\). We also use the fact that \(\nabla \tilde{u}\) is the projection of \(\nabla u^*\) onto the tangent space of the sphere, so that, provided \(|u^{n+1}-u^n|< 1/2\) holds, we have the point-wise estimate
1.3 Estimating \(r_{u,3}\)
The key to estimating \(r_{u,3}\) is to control \(\partial _t u^* \cdot u^*\). We notice that
and, thus,
so that it remains to understand \(I_1[\widehat{u} \times \widehat{\omega }]\cdot \widehat{u}\). Using orthogonality, we obtain
We insert (61) into (59) and obtain
Moreover, due to (55) we arrive at
Using (63) and (62) we obtain:
Let us note that for
we have suitable bounds for all terms on the right hand side of (65) except for \(\nabla \left( 1 - \frac{1}{|u^* |}\right) \), e.g., \(\nabla (I_1[\widehat{u}\times \widehat{\omega }] \cdot \widehat{u})\) can be estimated by applying the product rule to (61). As a first step towards estimating for \(\nabla \left( 1 - \frac{1}{|u^* |}\right) = - \frac{u^* \cdot \nabla u^*}{|u^* |^3}\), we compute
We recall \(\vert u^n\vert =1\), which implies \(\partial _{x_j} u^n \cdot u^n=0\), for all n, so that
We insert (67) into (66) and obtain
Thus, we obtain
This completes providing bounds for the different components of \(r_u\) and \(\nabla r_u\).
1.4 Estimating \(r_\omega \)
Obviously, \(\vert r_{\omega ,2}\vert = \vert a^\omega \vert \le \frac{1}{4} A^u A^u_{xx}\) and
We insert (54) and (53) into (70) and obtain
where we have used that
It remains to provide an estimate for \(|\Delta \tilde{u}- \Delta u^*\vert \). We note that \(u^*=\tilde{u}|u^* |\) and, thus,
so that
where we have used (68). Orthogonality \(\partial _{x_j} u^n \perp u^n\) for all j and all n implies
Inserting (74) into (73) implies
Inserting (75) into (71) completes the bound for \(r_\omega \).
1.5 Estimating \(r_g\)
We note that orthogonality \(u^n \perp \omega ^n\) implies
so that
Thus, using the definition of \(r_g\)
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Giesselmann, J., Mäder-Baumdicker, E. & Stonner, D.J. A posteriori error estimates for wave maps into spheres. Adv Comput Math 49, 54 (2023). https://doi.org/10.1007/s10444-023-10051-1
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DOI: https://doi.org/10.1007/s10444-023-10051-1