Abstract
We solve the variable coefficient Cauchy–Navier equations of elasticity in the unit square, for Dirichlet and Dirichlet-Neumann boundary conditions, using second order finite difference schemes. The resulting linear systems are solved by the preconditioned conjugate gradient (PCG) method with preconditioners corresponding to to the Laplace operator. The multiplication of a vector by the matrices of the resulting systems and the solution of systems with the preconditioners are performed at optimal and nearly optimal costs, respectively. For the case of Dirichlet boundary conditions, we prove the second order accuracy of the scheme in the discrete \(H^1\) norm, symmetry of the resulting matrix and its spectral equivalence to the preconditioner. For the case of Dirichlet–Neumann boundary conditions, we prove symmetry of the resulting matrix. Numerical tests demonstrating the convergence properties of the schemes and PCG are presented.

Similar content being viewed by others
References
Bialecki, B., Fairweather, G., Karageorghis, A.: Matrix decomposition algorithms for elliptic boundary value problems: a survey. Numer. Algoritm 56, 253–295 (2011)
Nečas, J., Hlaváček, I.: Mathematical Theory of Elastic and Elasto-plastic Bodies: An Introduction. Elsevier, Amsterdam–New York (1981)
Nilsson, S., Petersson, N.A., Sjögreen, B., Kreiss, H.-O.: Stable difference approximations for the elastic wave equation in second order formulation. SIAM J. Numer. Anal. 45, 1902–1936 (2007)
Samarskii, A.A., Andreev, W.B.: Difference Methods for Elliptic Equations. Nauka, Moscow (1976). in Russian
Sadd, M.H.: Elasticity Theory, Applications and Numerics. Academic Press, Elsevier, Amsterdam (2005)
Sadd, M.H.: Some simple Cartesian solutions to plane non-homogeneous elasticity problems. Mech. Res. Commun. 37, 22–27 (2010)
Samarskii, A.A., Nikolaev, E.S.: Numerical Methods for Grid Equations, vol. II: Iterative Methods. Birkhäuser Verlag, Basel (1989)
Sjögreen, B., Petersson, N.A.: A fourth order accurate finite difference scheme for the elastic wave equation in second order formulation. J. Sci. Comput. 52, 17–48 (2012)
MATLAB version 7.2.0.232 (R2006a), The MathWorks Inc., Natick, MA (2006)
Van Loan, C.: Computational Framework for the Fast Fourier Transform. Society for Industrial and Applied Mathematics, Philadelphia (1992)
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Bialecki, B., Karageorghis, A. Finite Difference Schemes for the Cauchy–Navier Equations of Elasticity with Variable Coefficients. J Sci Comput 62, 78–121 (2015). https://doi.org/10.1007/s10915-014-9847-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-014-9847-8