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Title: Rates of convergence in $W^2_p$-norm for the Monge-Amp\`ere equation.
We develop discrete $$W^2_p$$-norm error estimates for the Oliker--Prussner method applied to the Monge--AmpΓ¨re equation. This is obtained by extending discrete Alexandroff estimates and showing that the contact set of a nodal function contains information on its second-order difference. In addition, we show that the size of the complement of the contact set is controlled by the consistency of the method. Combining both observations, we show that the error estimate $$\| u - u_h \|_{W^2_{f,p}} (\mathcal{N}^I_h)$$ converges in order $$O(h^{1/p})$$ if $p > d$ and converges in order $$O(h^{1/d} \ln (\frac 1 h)^{1/d})$$ if $$p \leq d$$, where $$\|\cdot\|_{W^2_{f,p}(\mathcal{N}^I_h)}$$ is a weighted $$W^2_p$$-type norm, and the constant $C>0$ depends on $$\|{u}\|_{C^{3,1}(\bar\Omega)}$$, the dimension $$d$$, and the constant $$p$$. Numerical examples are given in two space dimensions and confirm that the estimate is sharp in several cases.  more » « less
Award ID(s):
1818861
PAR ID:
10101479
Author(s) / Creator(s):
;
Date Published:
Journal Name:
SIAM journal on numerical analysis
Volume:
56
Issue:
5
ISSN:
0036-1429
Page Range / eLocation ID:
3099–3120
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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