Abstract
The boundary element spline collocation method is studied for the time-fractional diffusion equation in a bounded two-dimensional domain. We represent the solution as the single layer potential which leads to a Volterra integral equation of the first kind. We discretize the boundary integral equation with the spline collocation method on uniform meshes both in spatial and time variables. In the stability analysis we utilize the Fourier analysis technique developed for anisotropic pseudodifferential equations. We prove that the collocation solution is quasi-optimal under some stability condition for the mesh parameters. We have to assume that the mesh parameter in time satisfies \((h_t=c h^{\frac{2}{\alpha}})\), where (h) is the spatial mesh parameter.
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References
Arnold, D., Saranen, J.: On the asymptotic convergence of spline collocation methods for partial differential equations. SIAM J. Numer. Anal. 21, 459–472 (1984)
Arnold, D.N., Wendland, W.L.: On the asymptotic convergence of collocation methods. Math. Comput. 41, 349–381 (1983)
Costabel, M.: Boundary integral operators for the heat equation. Int. Eq. Oper. Th. 13(4), 498–552 (1990)
Costabel, M., Saranen, J.: Spline collocation for convolutional parabolic boundary integral equations. Numer. Math. 84, 417–449 (2000)
Costabel, M., Saranen, J.: Parabolic boundary integral operators, symbolic representations and basic properties. Int. Eq. Oper. Th. 40, 185–211 (2001)
Hsiao, G., Saranen, J.: Coercivity of the Single Layer Heat Operator, Report 89-2. Center for Mathematics and Waves. Newark, Delaware (1989)
Hsiao, G.C., Saranen, J.: Boundary integral solution of the two-dimensional heat equation. Math. Methods Appl. Sci. 16, 87–114 (1993)
Hämäläinen, J.: Spline collocation for the single layer heat equation. Ann. Acad. Sci. Fenn. Math. Diss. 113, 67 pp. (1998)
Hörmander, L.: The analysis of linear partial differential operators I. Distribution Theory and Fourier Analysis. Springer-Verlag, Berlin (2003)
Kemppainen, J., Ruotsalainen, K.: Boundary integral solution of the time-fractional diffusion equation. Int. Eq. Oper. Th. 64, 239–249 (2009)
Kemppainen, J., Ruotsalainen, K.: Boundary element collocation method for time-fractional diffusion equation. In: Proc. of the 10th International Conference on Integral Methods in Science and Engineering. Birkhäuser, 7–10 July 2008 (2010)
Kilbas, A.A., Saigo, M.: H-transforms: Theory and Applications. CRC, LLC (2004)
Lions, J.L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications I. Springer, Berlin (1972)
Lions, J.L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications II. Springer, Berlin (1972)
Prudnikov, A.P., Brychkov, Y.A., Marichev, O.I.: Integrals and Series, vol. 3. More special functions. Overseas Publishers Association, Amsterdam (1990)
Saranen, J., Vainikko, G.: Periodic Integral and Pseudodifferential Equations with Numerical Approximations. Springer-Verlag, Berlin (2002)
Saranen, J., Wendland, W.L.: On the asymptotic convergence of collocation methods with spline functions of even degree. Math. Comput. 45, 91–108 (1985)
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Kemppainen, J.T., Ruotsalainen, K.M. On the spline collocation method for the single layer equation related to time-fractional diffusion. Numer Algor 57, 313–327 (2011). https://doi.org/10.1007/s11075-010-9430-9
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DOI: https://doi.org/10.1007/s11075-010-9430-9