Fast convergence of the Coiflet-Galerkin method for general elliptic BVPs

Akbari, H.

International Journal of Applied Mathematics and Computer Science, Tome 23 (2013) no. 1, pp. 17-27.

Résumé

We consider a general elliptic Robin boundary value problem. Using orthogonal Coifman wavelets (Coiflets) as basis functions in the Galerkin method, we prove that the rate of convergence of an approximate solution to the exact one is O(2^-nN) in the H¹ norm, where n is the level of approximation and N is the Coiflet degree. The Galerkin method needs to evaluate a lot of complicated integrals. We present a structured approach for fast and effective evaluation of these integrals via trivariate connection coefficients. Due to the fast convergence rate, very good approximations are found at low levels and with low Coiflet degrees, hence the size of corresponding linear systems is small. Numerical experiments confirm these claims.

Détail

Keywords: general stationary elliptic BVPs, Robin boundary condition, orthogonal Coifman wavelets, trivariate connection coefficients, fast convergence

Bibliographie

[1] Akbari, H. and Kerayechian, A. (2012). Coiflet-Galerkin method for solving second order BVPs with variable coefficients in three dimensions, Numerical Algorithms 61(4): 681–698, DOI: 10.1007/s11075-012-9558-x.

[2] Baccou, J. and Liandrat, J. (2006). Definition and analysis of a wavelet fictitious domain solver for the 2-D heat equation on a general domain, Mathematical Models and Methods in Applied Sciences 16(6): 819–845.

[3] Bandrowski, B., Karczewska, A. and Rozmej, P. (2010). Numerical solutions to integral equations equivalent to differential equations with fractional time, International Journal of Applied Mathematics and Computer Science 20(2): 261–269, DOI: 10.2478/v10006-010-0019-1.

[4] Cerna, D., Finek, V. and Najzar, K. (2008). On the exact values of coefficients of Coiflets, Central European Journal of Mathematics 6(1): 159–169.

[5] Daubechies, I. (1992). Ten Lectures on Wavelets, SIAM, Philadelphia, PA.

[6] El-Gamel, M. (2006). A wavelet-Galerkin method for a singularly perturbed convection-dominated diffusion equation, Applied Mathematics and Computation 181(2): 1635–1644.

[7] Ern, A. and Guermond, J. (2004). Theory and Practice of Finite Elements, Springer, New York, NY.

[8] Glowinski, R., Pan, T. W. and Periaux, J. (2006). Numerical simulation of a multi-store separation phenomenon: A fictitious domain approach, Computer Methods in Applied Mechanics and Engineering 195(41): 5566–5581.

[9] Hansen P. C. (1994). Regularization Tools: A Matlab package for analysis and solution of discrete Ill-posed problems, Numerical Algorithms 6: 1–35, http://www.mathworks.com/matlabcentral/fileexchange/52.

[10] Hashish, H., Behiry, S. H., Elsaid, A. (2009). Solving the 2-D heat equations using wavelet-Galerkin method with variable time step, Applied Mathematics and Computation 213(1): 209–215.

[11] Jensen, T. K. and Hansen, P. C. (2007). Iterative regularization with minimum-residual methods, BIT Numerical Mathematics 47(1): 103–120.

[12] Latto, A., Resnikoff, H. and Tenenbaum, E. (1992). The evaluation of connection coefficients of compactly supported wavele, Proceedings of the Workshop on Wavelets and Turbulence, Princeton, NJ, USA, pp. 76–89.

[13] Lin, E. and Zhou, X. (2001). Connection coefficients on an interval and wavelet solution of Burgers equation, Journal of Computational and Applied Mathematics 135(1): 63–78.

[14] Lin, E.and Zhou, X. (1997). Coiflet interpolation and approximate solutions of partial differential equations, Numerical Methods for Partial Differential Equations 13(4): 303–320.

[15] Nowak, Ł. D., Pasławska-Południak, M. and Twardowska, K. (2010). On the convergence of the wavelet-Galerkin method for nonlinear filtering, International Journal of Applied Mathematics and Computer Science 20(1): 93–108, DOI: 10.2478/v10006-010-0007-5.

[16] Reddy, J. (2006). An Introduction to the Finite Element Method, 3rd Edn., McGraw Hill, New York, NY.

[17] Resnikoff, H. and Wells, R. O. Jr (1998). Wavelet Analysis: The Scalable Structure of Information, Springer-Verlag, New York, NY.

[18] Romine, C. H. and Peyton, B. W. (1997). Computing connection coefficients of compactly supported wavelets on bounded intervals, Technical Report ORNL/TM-13413, Computer Science and Mathematical Division, Mathematical Sciences Section, Oak Ridge National Laboratory, Oak Ridge, TN, http://citeseer.ist.psu.edu/romine97computing.html.

[19] Saad, Y. (1996). Iterative Methods for Sparse Linear Systems, PWS Publishing Company.

[20] Saberi-Nadjafi, J., Mehrabinezhad, M. and Akbari, H. (2012). Solving Volterra integral equations of the second kind by wavelet-Galerkin scheme, Computers and Mathematics with Application 63(11): 1536–1547, DOI: 10.1016/j.camwa.2012.03.043.

[21] Vampa, V., Martin, M. and Serrano, E. (2010). A hybrid method using wavelets for the numerical solution of boundary value problems on the interval, Applied Mathematics and Computation 217(7): 3355–3367.