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Maximal $L^p$-regularity of time discretization and finite element methods

  • 演讲者:李步扬 (香港理工大学)

  • 时间:2017-12-22 15:30-16:00

  • 地点:慧园3栋 415报告厅

For a parabolic problem with maximal $L^p$-regularity, we prove that the time discretization by a linear multistep method or Runge-Kutta method preserves the maximal $L^p$-regularity uniformly in the stepsize if the method is A-stable (and satisfies minor additional conditions). In particular, the implicit Euler method, the Crank-Nicolson method, the second-order backward difference formula (BDF), and the Radau IIA and Gauss Runge-Kutta methods of all orders preserve maximal regularity. In general polygons and polyhedra, possibly nonconvex, the analyticity of the finite element heat semigroup in the $L^q$ norm, $1 \le q \le \infty$, and the maximal $L^p$-regularity of finite element solutions of parabolic equations are proved.