Abstract
We investigate a weak space-time formulation of the
heat equation and its use for the construction of a numerical scheme.
The formulation is based on a known weak space-time formulation, with
the difference that a pointwise component of the solution, which in
other works is usually neglected, is now kept. We investigate the role
of such a component by first using it to obtain a pointwise bound on
the solution and then deploying it to construct a numerical scheme.
The scheme obtained, besides being quasi-optimal in the
References
[1] Andreev R., Stability of space-time Petrov–Galerkin discretizations for parabolic evolution equations, PhD thesis, Dissertation no. 20842, ETH Zürich, 2012. Search in Google Scholar
[2] Andreev R., Stability of sparse space-time finite element discretizations of linear parabolic evolution equations, IMA J. Numer. Anal. 33 (2013), no. 1, 242–260. 10.1093/imanum/drs014Search in Google Scholar
[3] Andreev R., On long time integration of the heat equation, Calcolo 53 (2016), no. 1, 19–34. 10.1007/s10092-014-0133-9Search in Google Scholar
[4] Babuška I. and Aziz A. K., Survey lectures on the mathematical foundations of the finite element method, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (Baltimore 1972), Academic Press, New York (1972), 1–359. Search in Google Scholar
[5] Babuška I. and 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 (1989), no. 4, 363–399. 10.1002/num.1690050407Search in Google Scholar
[6] Babuška I. and 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 (1990), no. 4, 343–369. 10.1002/num.1690060406Search in Google Scholar
[7] N. Chegini and R. Stevenson , Adaptive wavelet schemes for parabolic problems: Sparse matrices and numerical results, SIAM J. Numer. Anal. 49 (2011), no. 1, 182–212. 10.1137/100800555Search in Google Scholar
[8] Cioica P. A., Dahlke S., Döhring N., Friedrich U., Kinzel S., Lindner F., Raasch T., Ritter K. and Schilling R. L., Convergence analysis of spatially adaptive Rothe methods, Found. Comput. Math. 14 (2014), no. 5, 863–912. 10.1007/s10208-013-9183-7Search in Google Scholar
[9] Ern A. and Guermond J., Theory and Practice of Finite Elements, Appl. Math. Sci. 159, Springer, New York, 2004. 10.1007/978-1-4757-4355-5Search in Google Scholar
[10] Larsson S. and Molteni M., A weak space-time formulation for the linear stochastic heat equation, Int. J. Appl. Comput. Math. (2016), 10.1007/s40819-016-0134-2. 10.1007/s40819-016-0134-2Search in Google Scholar
[11] Mollet C., Stability of Petrov–Galerkin discretizations: Application to the space-time weak formulation for parabolic evolution problems, Comput. Methods. Appl. Math. 14 (2013), no. 2, 231–255. 10.1515/cmam-2014-0001Search in Google Scholar
[12] Schwab C. and Stevenson R., Space-time adaptive wavelet methods for parabolic evolution problems, Math. Comp. 78 (2009), no. 267, 1293–1318. 10.1090/S0025-5718-08-02205-9Search in Google Scholar
[13] Schwab C. and Süli E., Adaptive Galerkin approximation algorithms for Kolmogorov equations in infinite dimensions, Stoch. PDE Anal. Comput. 1 (2013), no. 1, 483–493. 10.1007/s40072-013-0002-6Search in Google Scholar
[14] Tantardini F., Quasi-optimality in the backward Euler–Galerkin method for linear parabolic problems, PhD thesis, Università degli Studi di Milano, 2013. Search in Google Scholar
[15] Thomée V., Galerkin Finite Element Methods for Parabolic Problems, 2nd ed., Springer Ser. Comput. Math. 25, Springer, Berlin, 2006. Search in Google Scholar
[16] Urban K. and Patera A. T., An improved error bound for reduced basis approximation of linear parabolic problems, Math. Comp. 83 (2014), no. 288, 1599–1615. 10.1090/S0025-5718-2013-02782-2Search in Google Scholar
[17] Xu J. and Zikatanov L., Some observations on Babuška and Brezzi theories, Numer. Math. 94 (2003), no. 1, 195–202. 10.1007/s002110100308Search in Google Scholar
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