ERROR ANALYSIS OF AN IMMERSED FINITE ELEMENT METHOD FOR EULER-BERNOULLI BEAM INTERFACE PROBLEMS

This article presents an error analysis of a Hermite cubic immersed finite element (IFE) method for solving interface problems of the differential equation modeling a Euler-Bernoulli beam made up of multiple materials together with suitable jump conditions at material interfaces. The analysis consists of three essential groups. The first group is about IFE functions including bounds for the IFE shape functions and inverse inequalities. The second group is about error bounds for IFE interpolation derived with a multi-point Taylor expansion technique. The last group, and perhaps the most important group, is for proving the optimal convergence of the IFE solution generated by the usual Galerkin scheme based on the Hermite cubic IFE space considered in this article.

[1]  Philippe G. Ciarlet,et al.  Multipoint Taylor formulas and applications to the finite element method , 1971 .

[2]  I. Babuska,et al.  Generalized Finite Element Methods: Their Performance and Their Relation to Mixed Methods , 1983 .

[3]  I. Babuska,et al.  Finite Element Methods for the Solution of Problems with Rough Input Data. , 1985 .

[4]  Zhilin Li.  The immersed interface method using a finite element formulation , 1998 .

[5]  J. Stoer,et al.  Introduction to Numerical Analysis , 2002 .

[6]  Tao Lin,et al.  New Cartesian grid methods for interface problems using the finite element formulation , 2003, Numerische Mathematik.

[7]  Zhilin Li,et al.  An immersed finite element space and its approximation capability , 2004 .

[8]  Zhilin Li,et al.  An Immersed Finite Element Method for Elasticity Equations with Interfaces , 2005 .

[9]  Tzin Shaun Wang.  A Hermite Cubic Immersed Finite Element Space for Beam Design Problems , 2005 .

[10]  R. Kafafy,et al.  Three‐dimensional immersed finite element methods for electric field simulation in composite materials , 2005 .

[11]  Stefan A. Sauter,et al.  Composite Finite Elements for Elliptic Boundary Value Problems with Discontinuous Coefficients , 2006, Computing.

[12]  Gary R. Consolazio,et al.  Finite Elements , 2007, Handbook of Dynamic System Modeling.

[13]  Xiaoming He,et al.  Approximation capability of a bilinear immersed finite element space , 2008 .

[14]  Slimane Adjerid,et al.  A p-th degree immersed finite element for boundary value problems with discontinuous coefficients , 2009 .

[15]  Zhilin Li,et al.  Immersed Interface Finite Element Methods for Elasticity Interface Problems with Non-Homogeneous Jump Conditions , 2009 .

[16]  Théodore Papadopoulo,et al.  A Trilinear Immersed Finite Element Method for Solving the Electroencephalography Forward Problem , 2010, SIAM J. Sci. Comput..

[17]  P. Sharma.  Mechanics of materials. , 2010, Technology and health care : official journal of the European Society for Engineering and Medicine.

[18]  Z. Wang,et al.  Immersed finite element methods for 4th order differential equations , 2011, J. Comput. Appl. Math..

[19]  Tao Lin,et al.  Linear and bilinear immersed finite elements for planar elasticity interface problems , 2012, J. Comput. Appl. Math..

[20]  T. Lin,et al.  THE IMMERSED FINITE ELEMENT METHOD FOR PARABOLIC PROBLEMS USING THE LAPLACE TRANSFORMATION IN TIME DISCRETIZATION , 2013 .

[21]  Xu Zhang,et al.  A Method of Lines Based on Immersed Finite Elements for Parabolic Moving Interface Problems , 2013 .

[22]  Xiaoming He,et al.  Immersed finite element methods for parabolic equations with moving interface , 2013 .

[23]  Tao Lin,et al.  A locking-free immersed finite element method for planar elasticity interface problems , 2013, J. Comput. Phys..

[24]  Tao Lin,et al.  Partially Penalized Immersed Finite Element Methods For Elliptic Interface Problems , 2015, SIAM J. Numer. Anal..

[25]  Tao Lin,et al.  Partially penalized immersed finite element methods for parabolic interface problems , 2015, J. Comput. Appl. Math..

[26]  Slimane Adjerid,et al.  An immersed discontinuous finite element method for Stokes interface problems , 2015 .