×

A discontinuous Petrov-Galerkin method for time-fractional diffusion equations. (English) Zbl 1323.65109

The considered problem is as follows: to solve approximately \[ ^cD^{1-\alpha}u(x,t)-\Delta u(x,t)=f(x,t),\;\;(x,t)\in\Omega\times (0,T),\;\;u(x,0)=u_0(x), \] with homogeneous spatial Dirichlet conditions, where \(\Omega\subset \mathbb R^d\) is a convex polyhedral. \[ ^cD^{1-\alpha}v(t)=\int_0^t{{(t-s)^{\alpha-1}}\over{\Gamma(\alpha)}}v'(s)ds,\;\;0<\alpha<1, \] is the Caputo time-fractional derivative. The above problem can be met in physical, chemical, and biological applications. As it follows from the definition of \( ^cD^{1-\alpha}\), the time derivative at the point \(t\) depends on the ‘whole history’ from \(t=0\), hence step by step approximation with respect to time needs a special attention. For this reason the authors apply the discontinuous Petrov-Galerkin method for time variables, while the standard finite element method for spacial variables is used. The numerical scheme is based on piecewise polynomials of the degree \(r\geq1\). The authors present
a proof of well-posedness of the used method,
an error analysis (with large discussion),
advises concerning the choice of time steps near point \(t=0\),
two numerical examples with different smoothness.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
35K15 Initial value problems for second-order parabolic equations
35R11 Fractional partial differential equations
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
PDFBibTeX XMLCite
Full Text: DOI arXiv