A Variational Inequality Approach for One Dimensional Stefan Problem

Document Type : Original Article

Author
Mojtaba Moradipour
Department of Mathematics, Lorestan University, Khorramabad, Iran
10.29252/maco.1.2.4
Abstract
In this paper, we develop a numerical method to solve a famous free boundary PDE called the one dimensional Stefan problem.
First, we rewrite the PDE as a variational inequality problem (VIP). Using the linear finite element method, we discretize the variational inequality and achieve a linear complementarity problem (LCP). We present some existence and uniqueness theorems for solutions of the variational inequalities and free boundary problems. Finally we solve the LCP numerically by applying a modification of the active set strategy. 

Keywords

[1] M.H. AliAbadi and E.L. Ortiz, Numerical treatment of moving and free boundary value problems with the tau method, Computers & Mathematics with Applications, 35, 53 –61, 1998.
[2] R.W. Cottle, J.S. Pang and R.E. Stone, The Linear Complementarity Problem, Society for Industrial and Applied Mathematics, 2009.
[3] J. Crank, Free and moving boundary problems, Oxford science publications, Clarendon Press, 1984.
[4] Daniel J. Duffy. Finite difference methods in financial engineering. Wiley Finance Series. John Wiley & Sons, Ltd., Chichester, 2006. A partial differential equation approach, With 1 CD-ROM (Windows, Macintosh and UNIX).
[5] A. Friedman, Variational Principles and Free-Boundary Problems, Dover books on mathematics, Dover Publications, 2010.
[6] M. B. Giles and Rebecca Carter. Convergence analysis of crank-nicolson and rannacher time-marching. J. Comput. Finance, 9 (2006), 89–112.
[7] R. Glowinski. Numerical Methods for Nonlinear Variational Problems. Scientific Computation. Springer Berlin Heidelberg, 2013.
[8] A. D. Holmes. A Front-Fixing Finite Element Method for the Valuation of American Options. PhD thesis, University of Nevada, Las Vegas, 2010.
[9] S. Ikonen and J. Toivanen. Pricing American options using LU decomposition. Appl. Math. Sci. (Ruse), 1(49-52) (2007), 2529–2551.
[10] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applications, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000.
[11] M. Moradipour and S.A. Yousefi, Using spectral element method to solve variational inequalities with applications in finance, Chaos, Solitons & Fractals, 81 (2015), 208–217.
[12] J. L. Morales, J. Nocedal, and M. Smelyanskiy. An algorithm for the fast solution of symmetric linear complementarity problems. Numer. Math., 111(2) (2008), 251–266.
[13] R. Rannacher. Finite element solution of diffusion problems with irregular data. Numerische Mathematik, 43(2) (1984), 309–327.
[14] R. U. Seydel, Tools for computational finance, Universitext. Springer-Verlag, Berlin, fourth edition, 2009.
[15] P. Hartman and G. Stampacchia. On some non-linear elliptic differential-functional equations. Acta Mathematica, 115(1) (1966), 271–310.
[16] J. Toivanen. Numerical valuation of european and american options under kou’s jumpdiffusion model. SIAM Journal on Scientific Computing, 30(4) (2008), 1949–1970.
Mathematical Analysis and Convex Optimization
Volume 1, Issue 2
December 2020
Pages 35-43