We propose a new iterative scheme to compute the numerical solution to an over-determined boundary value problem for a general quasilinear elliptic PDE. The main idea is to repeatedly solve its linearization by using the quasi-reversibility method with a suitable Carleman weight function. The presence of the Carleman weight function allows us to employ a Carleman estimate to prove the convergence of the sequence generated by the iterative scheme above to the desired solution. The convergence of the iteration is fast at an exponential rate without the need of an initial good guess. We apply this method to compute solutions to some general quasilinear elliptic equations and a large class of first-order Hamilton-Jacobi equations. Numerical results are presented.
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The Carleman Contraction Mapping Method for Quasilinear Elliptic Equations with Over-determined Boundary Data
We propose a globally convergent numerical method to compute solutions to a general class
of quasi-linear PDEs with both Neumann and Dirichlet boundary conditions. Combining
the quasi-reversibility method and a suitable Carleman weight function, we define a map of
which fixed point is the solution to the PDE under consideration. To find this fixed point,
we define a recursive sequence with an arbitrary initial term using the same manner as in
the proof of the contraction principle. Applying a Carleman estimate, we show that the
sequence above converges to the desired solution. On the other hand, we also show that our
method delivers reliable solutions even when the given data are noisy. Numerical examples
are presented.
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- Award ID(s):
- 2208159
- NSF-PAR ID:
- 10407649
- Date Published:
- Journal Name:
- Acta Mathematica Vietnamica
- ISSN:
- 0251-4184
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s40306-023-00500-w
