Abstract
We propose a smooth fictitious domain/multiresolution method for enhancing the accuracy order in solving second order elliptic partial differential equations on general bivariate domains. We prove the existence and uniqueness of the solution of a corresponding discrete problem and a so-called interior error estimate which justifies the improved accuracy order. Numerical experiments are conducted on a Cassini oval.
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Angot, P., Bruneau, C., Fabrie, P.: A penalization method to take into account obstacles in incompressible viscous flow. Numer. Math. 81, 497–520 (1999)
Baccou, J., Liandrat, J.: On coupling wavelets with fictitious domain approaches. Appl. Math. Lett. 18, 1325–1331 (2005)
Baccou, J., Liandrat, J.: Definition and analysis of a wavelet/fictitious domain solver for the 2D-heat equation on a general domain. Math. Mod. Meth. Appl. S 16 (6), 819–845 (2006)
Benzi, M., Golub, G., Liesen, J.: Numerical solution of saddle point systems. Acta Numerica, 1–137 (2005)
Brezzi, F., Fortin, M.: Mixed and hybrid finite element methods. Springer (1991)
Burstedde, C., Kunoth, A.: A wavelet-based nested iteration-inexact conjugate gradient algorithm for adaptively solving elliptic PDEs. Numer. Algorithms 48, 161–188 (2008)
Chiavassa, G., Liandrat, J.: A fully adaptive wavelet algorithm for parabolic partial differential equations. Appl. Numer. Math. 36(2–3), 333–358 (2001)
Cohen, A.: Wavelets methods in numerical analysis. Handbook of numerical analysis. North-Holland (2000)
Cohen, A., Daubechies, I., Vial, P.: Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal. 1(1), 54–81 (1993)
Dahmen, W., Kunoth, A.: Appending boundary conditions by Lagrange multipliers: analysis of the LBB condition. Numer. Math. 88(1), 9–42 (2002)
Deriaz, E., Perrier, V.: Divergence-free and curl-free wavelets in two dimensions and three dimensions: application to turbulent flows. J. Turbul 7(3), 1–37 (2006)
Evans, L.C.: Partial differential equations. American Mathematical Society (1998)
Golub, G.H., Loan, C.F.V.: Matrix computations. The Johns Hopkins University Press (1996)
Haslinger, J., Kozubek, T., Kucera, R., Peichl, G.: Projected Schur complement method for solving non-symmetric systems arising from a smooth fictitious domain approach. Numer. Linear Algebra 14(9), 713–739 (2007)
Haslinger, J., Kozubek, T., Peichl, G.: A smooth embedding domain method based on the penalty approach. Num. Funct. Anal. Optim 32, 1252–1270 (2011)
Kunoth, A.: Wavelet techniques for the fictitious-domain-lagrange-multiplier-approach. Numer. Algorithms 27(3), 291–316 (2001)
Mommer, M.S.: Towards a fictitious domain method with optimally smooth solutions. Ph.D. thesis, RWTH-Aachen (2005)
Nicolaides, R.: Existence, uniqueness and approximation for generalized saddle point problems. SIAM J. Numer. Anal 19(2), 349–357 (1982)
Perrier, V., Basdevant, C.: La décomposition en ondelettes périodiques, un outil pour l’analyse de champs inhomogènes. théorie et algorithmes. La recherche aerospatiale 3, 57–67 (1989)
Quarteroni, A., Valli, A.: Numerical approximation of partial differential equations, Springer series in computational mathematics, vol. 23. Springer (1997)
Schneider, K., Farge, M.: Numerical simulation of the transient flow behaviour in tube bundles using a volume penalisation method. J. Fluid. Struct 20(4), 555–566 (2005)
Tomas, L.: Optimisation de forme et domaines fictifs: analyse de nouvelles formulations et aspects algorithmiques. Ph.D. thesis, Ecole centrale de Lyon (1997)
Yin, P., Liandrat, J.: Coupling wavelets/vaguelets and smooth fictitious domain methods for elliptic problems: the univariate case. Comput. Appl. Math. (2014). doi: 10.1007/s40314-014-0136-9
Yin, P., Liandrat, J., Shen, W.Q., Chen, Z.: A multiresolution and smooth fictitious domain method for one-dimensional elliptical and Stefan problems. Math. Comput. Model 58, 1727–1737 (2013)
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Yin, P., Liandrat, J. A smooth fictitious domain/multiresolution method for elliptic equations on general domains. Numer Algor 72, 705–720 (2016). https://doi.org/10.1007/s11075-015-0063-x
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DOI: https://doi.org/10.1007/s11075-015-0063-x