A nonconforming scheme with piecewise quasi three degree polynomial space to solve biharmonic problem
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A nonconforming scheme with piecewise quasi three degree polynomial space to solve biharmonic problem Shicang Song1 · Lijuan Lu1 Received: 18 June 2020 / Revised: 17 August 2020 / Accepted: 20 August 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract A new C 0 nonconforming quasi three degree element with 13 freedoms is introduced to solve biharmonic problem. The given finite element space consists of piecewise polynomial space P3 and some bubble functions. Different from non-C 0 nonconforming scheme, a smoother discrete solution can be obtained by this method. Compared with the existed 16 freedoms finite element method, this scheme uses less freedoms. As the finite elements are not affine equivalent each other, the associated interpolating error estimation is technically proved by introducing another affine finite elements. With this space to solve biharmonic problem, the convergence analysis is demonstrated between true solution and discrete solution. Under a stronger hypothesis that true solution u ∈ H02 () ∩ H 4 (), the scheme is of linear order convergence by the measurement of discrete norm · h . Some numerical results are included to further illustrate the convergence analysis. Keywords Biharmonic problem · Nonconforming element · Error estimation Mathematics Subject Classification 65N30
1 Introduction The biharmonic problem arises from many fields, such as fluid dynamics, plate bending problem, etc. Since it is not easy to find the exact solution, numerical methods, like finite element methods, become essential to obtain approximated solution.
This work is supported by the National Basic Research Program of China (Grant No. 2012CB025904).
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Shicang Song [email protected] Lijuan Lu [email protected] School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China
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S. Song, L. Lu
If one uses a conforming finite element method, which is X h ⊂ H02 (), to solve the biharmonic problem, the finite element space X h must satisfy X h ⊂ C 1 (). A method whose discrete space is contained in original finding space is frequently referred as a conforming method. To ensure that a piecewise polynomial vh ∈ X h is C 1 continuous or that mainly vh and its first-order partial derivatives are continuous across the common edge of two adjacent finite elements, it is necessary using at least two order derivatives as freedoms in X h . Several conforming methods [1] have been designed since 1970, such as Argyris triangle element with 21 freedoms, Bell triangle element with 18 freedoms and Boger–Fox–Schmit rectangular element with 16 freedoms. One can get smooth discrete solution by a conforming method, but it is always complex either to form nodal basis functions or construct stiffness matrixes using these conforming finite element schemes to solve biharmonic problem. To avoid complexities in conforming method, nonconforming elements have been developing in the last 3 or 4 decades. Various nonconforming elements are constructed in [1,7,10]. ACM (A
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