An ultraweak formulation of the Kirchhoff-Love plate bending model and DPG approximation
Authors:
Thomas Führer, Norbert Heuer and Antti H. Niemi
Journal:
Math. Comp. 88 (2019), 1587-1619
MSC (2010):
Primary 74S05, 74K20, 35J35; Secondary 65N30, 35J67
DOI:
https://doi.org/10.1090/mcom/3381
Published electronically:
October 5, 2018
MathSciNet review:
3925478
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Abstract | References | Similar Articles | Additional Information
Abstract: We develop and analyze an ultraweak variational formulation for a variant of the Kirchhoff-Love plate bending model. Based on this formulation, we introduce a discretization of the discontinuous Petrov-Galerkin type with optimal test functions (DPG). We prove well-posedness of the ultraweak formulation and quasi-optimal convergence of the DPG scheme.
The variational formulation and its analysis require tools that control traces and jumps in (standard Sobolev space of scalar functions) and
(symmetric tensor functions with
-components whose twice iterated divergence is in
), and their dualities. These tools are developed in two and three spatial dimensions. One specific result concerns localized traces in a dense subspace of
. They are essential to construct basis functions for an approximation of
.
To illustrate the theory we construct basis functions of the lowest order and perform numerical experiments for a smooth and a singular model solution. They confirm the expected convergence behavior of the DPG method both for uniform and adaptively refined meshes.
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Additional Information
Thomas Führer
Affiliation:
Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile
Email:
tofuhrer@mat.uc.cl
Norbert Heuer
Affiliation:
Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile
Email:
nheuer@mat.uc.cl
Antti H. Niemi
Affiliation:
Structures and Construction Technology Research Unit, Faculty of Technology, University of Oulu, Erkki Koiso-Kanttilan katu 5, Linnanmaa, 90570 Oulu, Finland
Email:
antti.niemi@oulu.fi
DOI:
https://doi.org/10.1090/mcom/3381
Keywords:
Kirchhoff--Love model,
plate bending,
biharmonic problem,
fourth-order elliptic PDE,
discontinuous Petrov--Galerkin method,
optimal test functions
Received by editor(s):
September 22, 2017
Received by editor(s) in revised form:
March 23, 2018, and May 11, 2018
Published electronically:
October 5, 2018
Additional Notes:
This research was supported by CONICYT through FONDECYT projects 1150056, 11170050, The Magnus Ehrnrooth Foundation, and by Oulun rakennustekniikan säätiö.
The second author is the corresponding author.
Article copyright:
© Copyright 2018
American Mathematical Society