Matrix equation solving of PDEs in polygonal domains using conformal mappings

References

[1] R. H. Bartels and G. W. Stewart, Solution of the matrix equation AX + XB = C, Commun. ACM, 15 (1972), 820–826.10.1145/361573.361582Search in Google Scholar

[2] M. Z. Bazant, Conformal mapping of some non-harmonic functions in transport theory, Proc. R. Soc. Lond. A, 460 (2004), 1433–1452.10.1098/rspa.2003.1218Search in Google Scholar

[3] P. Benner and T. Damm, Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems, SIAM J. Control Optim., 49 (2011), 686–711.10.1137/09075041XSearch in Google Scholar

[4] P. Benner, J.-R. Li, and T. Penzl, Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems, Numer. Linear Alg. Appl., 15 (2008), 1–23.10.1002/nla.622Search in Google Scholar

[5] M. Benzi, Preconditioning techniques for large linear systems: a survey, J. Comput. Phys., 182 (2002), 418–477.10.1006/jcph.2002.7176Search in Google Scholar

[6] C. Canuto, V. Simoncini, and M. Verani, Contraction and optimality properties of an adaptive Legendre–Galerkin method: the multi-dimensional case, J. Sci. Comput., 63 (2014), 769–798.10.1007/s10915-014-9912-3Search in Google Scholar

[7] J. E. Castillo, ed., Mathematical Aspects of Numerical Grid Generation, Frontiers in Applied Mathematics, SIAM, 1991.10.1137/1.9781611971019Search in Google Scholar

[8] S. Chakravarthy and D. Anderson, Numerical conformal mapping, Math. Comput., 33 (1979), 953–969.10.1090/S0025-5718-1979-0528049-6Search in Google Scholar

[9] N. V. Challis and D. M. Burley, A numerical method for conformal mapping, IMA J. Numer. Analysis, 2 (1982), 169–181.10.1093/imanum/2.2.169Search in Google Scholar

[10] T. Damm, Direct methods and ADI-preconditioned Krylov subspace methods for generalized Lyapunov equations, Num. Linear Alg. Appl., 15 (2008), 853–871.10.1002/nla.603Search in Google Scholar

[11] T. Damm, P. Benner, and J. Hauth, Computing the stochastic H^∞-norm by a Newton iteration, IEEE Control Systems Letters, 1 (2017), No. 1, 92–97.10.1109/LCSYS.2017.2707409Search in Google Scholar

[12] T. Damm, K. Sato, and A. Vierling, Numerical solution of Lyapunov equations related to Markov jump linear systems, Numer Linear Alg. Appl., 25 (2017), No. 6, e2113.10.1002/nla.2113Search in Google Scholar

[13] T. A. Driscoll, Algorithm 756: A MATLAB toolbox for Schwarz–Christoffel mapping, ACM Trans. Math. Softw., 22 (1996), 168–186.10.1145/229473.229475Search in Google Scholar

[14] T. A. Driscoll, Algorithm 843: Improvements to the Schwarz–Christoffel toolbox for MATLAB, ACM Trans. Math. Softw., 31 (2005), 239–251.10.1145/1067967.1067971Search in Google Scholar

[15] T. A. Driscoll and L. N. Trefethen, Schwarz–Christoffel Mapping, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, 2002.10.1017/CBO9780511546808Search in Google Scholar

[16] V. Druskin and V. Simoncini, Adaptive rational Krylov subspaces for large-scale dynamical systems, Systems Control Letters, 60 (2011), 546–560.10.1016/j.sysconle.2011.04.013Search in Google Scholar

[17] N. S. Ellner and E. L. Wachspress, New ADI model problem applications, In: Proc. of 1986 ACM Fall Joint Computer Conference, Dallas, Texas, United States, IEEE Computer Society Press, Los Alamitos, CA, 1986, pp. 528–534.Search in Google Scholar

[18] H. C. Elman, D. J. Silvester, and A. J. Wathen, Finite Elements and Fast Iterative Solvers, with Applications in Incompressible Fluid Dynamics, 2nd ed., Oxford University Press, Oxford, 2014.10.1093/acprof:oso/9780199678792.001.0001Search in Google Scholar

[19] M. Farrashkhalvat and J. P. Miles, Basic Structured Grid Generation, Butterworth & Heinemann, 2003.10.1016/B978-075065058-8/50008-3Search in Google Scholar

[20] B. Fornberg, A numerical method for conformal mappings, SIAM J. Sci. Stat. Comp., 1 (1980), 386–400.10.1137/0901027Search in Google Scholar

[21] G. Golub and C. F. Van Loan, Matrix Computations, The Johns Hopkins University Press, Baltimore, 4th ed., 2013.Search in Google Scholar

[22] L. H. Howell and L. T. Trefethen, A modified Schwarz–Christoffel transformation for elongated regions, SIAM J. Sci. Stat. Comput., 11, (1990), No. 5, 928–949.10.1137/0911054Search in Google Scholar

[23] D. C. Ives, Conformal grid generation, Appl. Math. Comput., 10-11 (1982), 107–135.10.1016/0096-3003(82)90189-8Search in Google Scholar

[24] E. Jarlebring, G. Mele, D. Palitta, and E. Ringh, Krylov methods for low-rank commuting generalized Sylvester equations, Numer. Linear Alg. Appl., 25 (2018), No. 6, e2176.10.1002/nla.2176Search in Google Scholar

[25] P. M. Knupp and S. Steinberg, The Fundamentals of Grid Generation, Knupp, 1992.Search in Google Scholar

[26] C. W. Mastin and J. F. Thompson, Quasiconformal mappings and grid generation, SIAM J. Sci. Stat. Comp., 5 (1984), 305–310.10.1137/0905022Search in Google Scholar

[27] The MathWorks, Inc., MATLAB 7, r2017b ed., 2017.Search in Google Scholar

[28] M. A. Olshanskii and E. E. Tyrtyshnikov, Iterative Methods for Linear Systems, Theory and Applications, SIAM, 2014.10.1137/1.9781611973464Search in Google Scholar

[29] D. Palitta and V. Simoncini, Matrix-equation-based strategies for convection–diffusion equations, BIT Numer. Math., 56 (2016), 751–776.10.1007/s10543-015-0575-8Search in Google Scholar

[30] D. Palitta and V. Simoncini, Optimality properties of Galerkin and Petrov–Galerkin methods for linear matrix equations, Vietnam J. Math., 48 (2020), 791–807.10.1007/s10013-020-00390-7Search in Google Scholar

[31] C. E. Powell, D. Silvester, and V. Simoncini, An efficient reduced basis solver for stochastic galerkin matrix equations, SIAM J. Sci. Comp., 39 (2017), A141–A163.10.1137/15M1032399Search in Google Scholar

[32] Y. Saad, Iterative Methods for Sparse Linear Systems, Society for Industrial and Applied Mathematics, 2nd ed., 2003.10.1137/1.9780898718003Search in Google Scholar

[33] G. Sangalli and M. Tani, Isogeometric preconditioners based on fast solvers for the Sylvester equation, SIAM J. Sci. Comput., 38 (2016), A3644–A3671.10.1137/16M1062788Search in Google Scholar

[34] S. D. Shank, V. Simoncini, and D. B. Szyld, Efficient low-rank solutions of generalized Lyapunov equations, Numerische Mathematik, 134 (2016), 327–342.10.1007/s00211-015-0777-7Search in Google Scholar

[35] V. Simoncini, Computational methods for linear matrix equations, SIAM Review, 58 (2016), 377–441.10.1137/130912839Search in Google Scholar

[36] D. Sorensen and Y. Zhou, Direct methods for matrix Sylvester and Lyapunov equations, J. Appl. Math., 2003 (2003), No. 6, 277–303.10.1155/S1110757X03212055Search in Google Scholar

[37] G. T. Symm, An integral equation method in conformal mapping, Numer. Math., 9 (1966), 250–258.10.1007/BF02162088Search in Google Scholar

[38] J. F. Thompson, Numerical Grid Generation, Elsevier, Amsterdam, 1982.10.1016/0096-3003(82)90188-6Search in Google Scholar

[39] J. F. Thompson, Z. U. Warsi, and C. W. Mastin, Numerical Grid Generation: Foundations and Applications, Elsevier, North-Holland, Inc., USA, 1985.Search in Google Scholar

[40] L. N. Trefethen, Numerical computation of the Schwarz–Christoffel transformation, SIAM J. Sci. Comput., 1 (1980), 82–102.10.1137/0901004Search in Google Scholar

[41] L. N. Trefethen, Numerical conformal mapping with rational functions, Comput. Methods Function Theory 20 (2020), No. 3, 369–387.10.1007/s40315-020-00325-wSearch in Google Scholar

[42] E. L. Wachspress, Iterative Solution of Elliptic Systems, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1966.Search in Google Scholar