Excerpt from High Order Fast Laplace Solvers for the Dirichlet Problem on General Regions
Numerical experiments, see Pereyra [13] and the last section of this paper, clearly demonstrate the need for higher order accuracy at the irregular mesh points if improved solutions through Richardson extrapolation or deferred correction methods are required. In his 1966 paper, Pereyra also reported on successful numerical experiments with methods based on Lagrange interpolation in one variable and employing only mesh points close to the boundary. At that time no convergence proof was known for such methods.
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