We develop a new finite element method for solving planar elasticity problems involving heterogeneous materials with a mesh not necessarily aligning with the interface of the materials. This method is based on the ‘broken’ Crouzeix–Raviart -nonconforming finite element method for elliptic interface problems [D.Y. Kwak, K.T. Wee and K.S. Chang, SIAM J. Numer. Anal. 48 (2010) 2117–2134]. To ensure the coercivity of the bilinear form arising from using the nonconforming finite elements, we add stabilizing terms as in the discontinuous Galerkin (DG) method [D.N. Arnold, SIAM J. Numer. Anal. 19 (1982) 742–760; D.N. Arnold and F. Brezzi, in Discontinuous Galerkin Methods. Theory, Computation and Applications, edited by B. Cockburn, G.E. Karniadakis, and C.-W. Shu. Vol. 11 of Lecture Notes in Comput. Sci. Engrg. Springer-Verlag, New York (2000) 89–101; M.F. Wheeler, SIAM J. Numer. Anal. 15 (1978) 152–161.]. The novelty of our method is that we use meshes independent of the interface, so that the interface may cut through the elements. Instead, we modify the basis functions so that they satisfy the Laplace–Young condition along the interface of each element. We prove optimal and divergence norm error estimates. Numerical experiments are carried out to demonstrate that our method is optimal for various Lamè parameters and and locking free as .
Mots clés : Immersed finite element method, Crouzeix–Raviart finite element, elasticity problems, heterogeneous materials, stability terms, Laplace–Young condition
@article{M2AN_2017__51_1_187_0, author = {Kwak, Do Y. and Jin, Sangwon and Kyeong, Daehyeon}, title = {A stabilized $P_{1}$-nonconforming immersed finite element method for the interface elasticity problems}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {187--207}, publisher = {EDP-Sciences}, volume = {51}, number = {1}, year = {2017}, doi = {10.1051/m2an/2016011}, zbl = {1381.74199}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2016011/} }
TY - JOUR AU - Kwak, Do Y. AU - Jin, Sangwon AU - Kyeong, Daehyeon TI - A stabilized $P_{1}$-nonconforming immersed finite element method for the interface elasticity problems JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2017 SP - 187 EP - 207 VL - 51 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2016011/ DO - 10.1051/m2an/2016011 LA - en ID - M2AN_2017__51_1_187_0 ER -
%0 Journal Article %A Kwak, Do Y. %A Jin, Sangwon %A Kyeong, Daehyeon %T A stabilized $P_{1}$-nonconforming immersed finite element method for the interface elasticity problems %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2017 %P 187-207 %V 51 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2016011/ %R 10.1051/m2an/2016011 %G en %F M2AN_2017__51_1_187_0
Kwak, Do Y.; Jin, Sangwon; Kyeong, Daehyeon. A stabilized $P_{1}$-nonconforming immersed finite element method for the interface elasticity problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 1, pp. 187-207. doi : 10.1051/m2an/2016011. http://www.numdam.org/articles/10.1051/m2an/2016011/
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