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Solving the biharmonic Dirichlet problem on domains with corners
Author(s) -
De Coster Colette,
Nicaise Serge,
Sweers Guido
Publication year - 2015
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201400022
Subject(s) - biharmonic equation , mathematics , dirichlet problem , uniqueness , constructive , domain (mathematical analysis) , bounded function , focus (optics) , boundary value problem , order (exchange) , dirichlet distribution , mathematical analysis , computer science , physics , process (computing) , finance , operating system , optics , economics
The biharmonic Dirichlet boundary value problem on a bounded domain is the focus of the present paper. By Riesz' representation theorem the existence and uniqueness of a weak solution is quite direct. The problem that we are interested in appears when one is looking for constructive approximations of a solution. Numerical methods using for example finite elements, prefer systems of second equations to fourth order problems. Ciarlet and Raviart in [7][P. G. Ciarlet, 1974] and Monk in [21][P. Monk, 1987] consider approaches through second order problems assuming that the domain is smooth. We will discuss what happens when the domain has corners. Moreover, we will suggest a setting, which is in some sense between Ciarlet‐Raviart and Monk, that inherits the benefits of both settings and that will give the weak solution through a system type approach.