Abstract
An algorithm for a stable parallelizable space-time Petrov-Galerkin discretization for linear parabolic evolution equations is given. Emphasis is on the reusability of spatial finite element codes.
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Akrivis, G., Makridakis, C., Nochetto, R.: Galerkin and Runge–Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence. Numer. Math. 118, 429–456 (2011)
Alberty, J., Carstensen, C., Funken, S.A.: Remarks around 50 lines of Matlab: short finite element implementation. Numer. Algorithm. 20(2-3), 117–137 (1999)
Andreev, R.: Stability of space-time Petrov-Galerkin discretizations for parabolic evolution equations. Ph.D. thesis. ETH Zu¨rich, ETH Diss. No. 20842 (2012)
Andreev, R.: Stability of sparse space-time finite element discretizations of linear parabolic evolution equations. IMA J. Numer. Anal. 33(1), 242–260 (2013)
Andreev, R., Schweitzer, J.: Conditional space-time stability of collocation Runge–Kutta for parabolic evolution equations. Tech. Rep. 2013-15 (2013)
Andreev, R., Tobler, C.: Multilevel preconditioning and low rank tensor iteration for space-time simultaneous discretizations of parabolic PDEs. Tech. Rep. ETH Zürich (2012)
Babuška, I., Janik, T.: The h-p version of the finite element method for parabolic equations. I. The p Version in Time. Numer. Methods Partial Differential Equations 5, 363–399 (1989)
Babuška, I., Janik, T.: The h-p version of the finite element method for parabolic equations. II. The h-p Version in Time Numer. Methods Partial Differential Equations 6, 343–369 (1990)
Banjai, L.: Parallel multistep methods for linear evolution problems. IMA J. Numer. Anal. 32(3), 1217–1240 (2012)
Benbow, S.J.: Solving generalized least-squares problems with LSQR. SIAM J. Matrix Anal. Appl. 21(1), 166–177 (1999)
Chang, X.W., Paige, C.C., Titley-Peloquin, D.: Stopping criteria for the iterative solution of linear least squares problems. SIAM J. Matrix Anal. Appl. 31(2), 831–852 (2009)
Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics, vol. 19. J American Mathematical Society (1998)
Fattorini, H.O.: Infinite dimensional linear control systems, North-Holland Mathematics Studies, vol. 201. Elsevier Science B.V., Amsterdam (2005)
Hulme, B.L.: One-step piecewise polynomial Galerkin methods for initial value problems. Math. Comp. 26, 415–426 (1972)
Lions, J.L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications, vol. I. Springer-Verlag, New York (1972)
Paige, C.C., Saunders, M.A.: LSQR: an algorithm for sparse linear equations and sparse least squares. ACM Trans. Math. Software 8(1), 43–71 (1982)
Schwab, C., Stevenson, R.: Space-time adaptive wavelet methods for parabolic evolution problems. Math. Comp. 78(267), 1293–1318 (2009)
Sheen, D., Sloan, I.H., Thom´ee, V.: A parallel method for time discretization of parabolic equations based on Laplace transformation and quadrature. IMA J. Numer. Anal 23(2), 269–299 (2003)
Xu, J., Zikatanov, L.: Some observations on Babuˇska and Brezzi theories. Numer. Math 94(1), 195–202 (2003)
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Andreev, R. Space-time discretization of the heat equation. Numer Algor 67, 713–731 (2014). https://doi.org/10.1007/s11075-013-9818-4
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DOI: https://doi.org/10.1007/s11075-013-9818-4