A posteriori error analysis and adaptive schemes for the wave equation

Aposteriori error estimates provide a rigorous foundation for the derivation of efficient adaptive

algorithms for the approximation of solutions of partial differential equations (PDEs). While the

literature is rich with results for the approximation of elliptic and parabolic PDEs, it is much less

developed for the hyperbolic equations such as the acoustic or elastic wave equations. In this

talk, I will review some of the “standard” aposteriori results for the scalar linear wave equation,

including those of [1] and [2], and present recent improvements and further developments to

lower order Sobolev norms based on Baker’s Trick [3] for backward Euler schemes.

Subsequent focus will be given to practically relevant methods such as Verlet, Cosine, or

Newmark methods, a popular example of which is the Leap-frog method [4].

Notes: This is based on joint work with E.H. Georgoulis, C. Makridakis and J.M. Virtanen.

References:

[1] W. Bangerth and R. Rannacher, J. Comput. Acoust. 9(2):575–591, 2001.

[2] C. Bernardi and E. Süli, Math. Models Methods Appl. Sci. 15(2):199–225, 2005.

[3] E. H. Georgoulis, O. Lakkis, and C. Makridakis. IMA J. Numer. Anal., 33(4):1245–1264,

2013, http://arxiv.org/abs/1003.3641

[4] E. H. Georgoulis, O. Lakkis, C. Makridakis, and J. M. Virtanen. SIAM J. Numer. Anal.,

54(1), 2016, http://arxiv.org/abs/1411.7572