|
|
Solving the 2-D Elliptic Monge-Ampère Equation by a Kansa's Method |
Qin LI1, Zhi-yong LIU2 |
1 School of Science, Beijing Technology and Business University, Beijing 100048, China; 2 School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China |
|
|
Abstract In this paper, a Kansa's method is designed to solve numerically the Monge-Ampère equation. The primitive Kansa's method is a meshfree method which applying the combination of some radial basis functions (such as Hardy's MQ) to approximate the solution of the linear parabolic, hyperbolic and elliptic problems. But this method is deteriorated when is used to solve nonlinear partial differential equations. We approximate the solution in some local triangular subdomains by using the combination of some cubic polynomials. Then the given problems can be computed in each subdomains independently. We prove the stability and convergence of the new method for the elliptic Monge-Ampère equation. Finally, some numerical experiments are presented to demonstrate the theoretical results.
|
Received: 11 June 2015
|
|
Fund:The first author is supported in part by the National Natural Science Foundations of China (No.11426039, 11571023, 11471329). The second author is partially supported by the National Natural Science Foundation of China (No.11501313), the Natural Science Foundation of Ningxia Province (No.NZ15005), and the Science Research Project of Ningxia Higher Education (No.NGY2016059). |
About author:: 65N12;65N99;35J60 |
|
|
|
[1] |
Barles, G., Souganidis, P.E. Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal., 4:271-283(1991)
|
[2] |
Benamou, J.D., Froese, B.D., Oberman A.M. Two numerical methods for the elliptic Monge-Ampère equation. ESAIM:Math. Model. Numer. Anal., 44:737-758(2010)
|
[3] |
Böhmer, K. Numerical Methods for Nonlinear Elliptic Differential Equations:a Synopsis. Oxford University Press, New York, 2010
|
[4] |
Brenner, S.C., Gudi, T., Neilan, M., Sung, L.Y. C0 penalty methods for the fully nonlinear Monge-Ampère equation. Math. Comput., 80(276):1979-1995(2011)
|
[5] |
Brenner, S.C., Scott, L.R. The Mathematical Theory of Finite Element Methods, 3nd Edition. SpringerVerlag, New York, 2008
|
[6] |
Ciarlet, P.G. The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam, 1978
|
[7] |
Dean, E.J., Glowinski, R. An augmented Lagrangian approach to the numerical solution of the Dirichlet problem for the elliptic Monge-Ampère equation in two dimensions. Electron. Trans. Numer. Anal., 22:71-96(2006)
|
[8] |
Dean, E.J., Glowinski, R. Numerical methods for fully nonlinear elliptic equations of the Monge-Ampère type. Comput. Methods Appl. Mech. Engrg, 195:1344-1386(2006)
|
[9] |
Dean, E.J., Glowinski, R. On the numerical solution of the elliptic Monge-Ampère equation in dimension two:a leastsquares approach. In:Partial Differential Equations, Comput. Methods Appl. Sci., Vol.16, Springer, Dordrecht, the Netherlands, 2008, 43-63
|
[10] |
Dean, E.J., Glowinski, R., Pan, T.-W. Operator-splitting methods and applications to the direct numerical simulation of particulate flow and to the solution of the elliptic Monge-Ampère equation. In:Control and boundary analysis, Lect. Notes Pure Appl. Math., Vol.240, Chapman & Hall/CRC, Boca Raton, USA, 1-27(2005)
|
[11] |
Dennis Jr, J.E., Schnabel, R.B. Numerical Methods for Unconstrained Optimization and Nonlinear Equations. SIAM, Philadelphia, 1996
|
[12] |
Fasshauer, G.E. Meshfree Approximation Methods with MATLAB. World Scientific Publishers, Singapore, 2007
|
[13] |
Fedoseyev, A.I., Friedman, M.J., Kansa, E.J. Continuation for nonlinear elliptic partial differential equations discretized by the multiquadric method. Int. J. Bifurcation and Chaos, 10(2):481-492(2000)
|
[14] |
Feng, X., Neilan, M. Mixed finite element methods for the fully nonlinear Monge-Ampère equation based on the vanishing moment method. SIAM J. Numer. Anal., 47:1226-1250(2009)
|
[15] |
Feng, X., Neilan, M. Analysis of Galerkin methods for the fully nonlinear Monge-Ampère equation. J. Sci. Comput., 47:303-327(2011)
|
[16] |
Froese, B.D., Oberman, A.M. Fast finite difference solvers for singular solutions of the elliptic MongeAmpère equation. J. Comput. Phys., 230:818-834(2011)
|
[17] |
Froese, B.D., Oberman, A.M. Convergent finite difference solvers for viscosity solutions of the elliptic Monge-Ampère equation in dimensions two and higher. SIAM J. Numer. Anal., 49(4):1692-1714(2012)
|
[18] |
Glowinski, R. Numerical methods for fully nonlinear elliptic equations. In:6th International Congress on Industrial and Applied Mathematics, ICIAM 07, Invited Lectures, R. Jeltsch and G. Wanner Eds., 2009, 155-192
|
[19] |
Glowinski, R., Dean, E.J., Guidoboni, G., Juárez, L.H., Pan, T.-W. Applications of operator-splitting methods to the direct numerical simulation of particulate and free-surface flows and to the numerical solution of the two-dimensional elliptic Monge-Ampère equation. Japan J. Indust. Appl. Math., 25:1-63(2008)
|
[20] |
Haltiner, G.J. Numerical Weather Prediction. Wiley, New York, 1971
|
[21] |
Islam, S., Haq, S., Uddin, M. A meshfree interpolation method for the numerical solution of the coupled nonlinear partial differential equations. Engineering Analysis with Boundary Elements, 33:399-409(2009)
|
[22] |
Kansa E.J. Application of Hardy's multiquadric interpolation to hydrodynamics, In:Proceedings of the 1986 Annual Simulations Conference, Vol. 4, San Diego, CA, 1986, 111-117
|
[23] |
Kansa, E.J. Multiquadrics-a scattered data approximation scheme with applications to computational fluid-dynamics-Ⅱ. Computers Math. Applic, 19(8/9):147-161(1990)
|
[24] |
Kasahara, A. Significance of non-elliptic regions in balanced flows of the tropical atmosphere. Mon. Weather Rev, 110:1956-1967(1982)
|
[25] |
Kelley, C.T. Solving Nonlinear Equations with Newton's Method. SIAM, Philadelphia, 2003
|
[26] |
Khattak, A.J., Tirmizi, S.I.A., Islam, S. Application of meshfree collocation method to a class of nonlinear partial differential equations. Engineering Analysis with Boundary Elements, 33:661-667(2009)
|
[27] |
Oberman, A.M. Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian. Discrete Contin. Dyn. Syst. Ser. B, 10:221-238(2008)
|
[28] |
Oliker, V.I., Prussner, L.D. On the numerical solution of the equation (∂2z)/(∂x2) (∂2z)/(∂y2)-((∂2z)/(∂x∂y))2=f and its discretizations, I. Numer. Math., 54:271-293(1988)
|
[29] |
Stojanovic, S.D. Risk premium and fair option prices under stochastic volatility:the hara solution. C. R. Math, 340:551-556(2005)
|
[30] |
Stoker, J.J. Nonlinear Elasticity. Gordon and Breach Science Publishers, New York, 1968
|
[31] |
Wendland, H. Piecewise polynomial, positive definite and compactly supproted radial functions of minimal degree. Adv. Comput. Math., 4:389-396(1995)
|
[32] |
Westcott, B.S. Shaped Reflector Antenna Design. Research Studies Press, New York, 1983
|
|
|
|