A mollification method for a Cauchy problem for the Laplace equation
type: Item
instance of: scholarly article
title: A mollification method for a Cauchy problem for the Laplace equation
zbMATH Open document ID: 1221.65249
published in: Applied Mathematics and Computation
publication date: 2011-06-27T00:00:00Z
review text: 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)].
reviewed by: Dinh Nho Hào
Mathematics Subject Classification ID: 65M30
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 65M38
zbMATH DE Number: 5913733
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Laplace equation
zbMATH Keywords: ill-posed problem
zbMATH Keywords: regularization
zbMATH Keywords: mollification method
zbMATH Keywords: error estimate
zbMATH Keywords: numerical examples
author: Zhen-Ping Li
author: Chuli Fu
MaRDI profile type: Publication
full work available at URL: j.amc.2011.03.160
OpenAlex ID: W2121690246
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DOI: 10.1016/J.AMC.2011.03.160
description: scientific article; zbMATH DE number 5913733