Applied Analysis Seminar - Abstracts
Talk
Tuesday, 4 February 2020, 13:30-14:30 in HG02.802
Lehel Banjai (Heriot-Watt University)
A tensor finite element method for a space fractional wave equation.
Abstract
We study solution techniques for an evolution equation involving second order derivative
in time and the spectral fractional powers of symmetric, coercive, linear, elliptic,
second-order operators in bounded spatial domains. We realize fractional diffusion as
the Dirichlet-to-Neumann map for a nonuniformly elliptic problem posed on a semi-infinite
cylinder. We thus rewrite our evolution problem as a quasi-stationary elliptic problem
with a dynamic boundary condition and derive space, time, and space-time regularity
estimates for its solution. The latter problem exhibits an exponential decay in the
extended dimension and thus suggests a truncation that is suitable for numerical
approximation. We propose and analyze two fully discrete schemes. The discretization
in time is based on finite difference discretization techniques: trapezoidal and leapfrog
schemes. The discretization in space relies on the tensorization of a first-degree FEM in
with a suitable hp-FEM in the extended variable. For both schemes we derive stability and
error estimates and present numerical results.
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