A mollification method for a Cauchy problem for the Laplace equation
The authors consider the Cauchy problem for the Laplace equation in an infinite strip \(u_{xx} + u_{yy} =0, x \in (0,1), y \in \mathbb{R}, u(0,y) = \varphi(y), u_x(0,y) = 0, y \in \mathbb{R}\) with noisy Cauchy data \(\varphi^\delta \in L^2(\mathbb{R})\). They mollify the Cauchy data by the convolution with the Gaussian kernel and establish error estimates between the exact and mollified solutions for a priori chosen mollification parameters. They discuss also a posteriori choices of the mollification parameters. Some numerical examples are presented. Note that the mollification method by convolution with the Gaussian kernel for the Cauchy problem for the Laplace equation appeared much earlier in the work by \textit{V. K.\ Ivanov} [Differ. Equations 1, 98--102 (1965); translation from Differ. Uravn. 1, 131--136 (1965; Zbl 0192.20101)]. Afterwards, \textit{V. V. Vasin} applied this method to numerical differentiation [U.S.S.R. Comput. Math. Math. Phys. 13, No.~6, 16--24 (1973); translation from Zh. Vychisl. Mat. Mat. Fiz. 13, 1383--1389 (1973; Zbl 0306.65007)], and his work has been somewhat improved by \textit{C. W. Groetsch} [ J. Optimization Theory Appl. 74, No.~2, 373--378 (1992; Zbl 0795.90063)].
scientific article; zbMATH DE number 5913733