Bishnu Lamichhane

Jérôme Droniou, Muhammad Ilyas, Bishnu Lamichhane et al.

We consider a saddle point formulation for a sixth order partial differential equation and its finite element approximation, for two sets of boundary conditions. We follow the Ciarlet-Raviart formulation for the biharmonic problem to formulate our saddle point problem and the finite element method. The new formulation allows us to use the $H^1$-conforming Lagrange finite element spaces to approximate the solution. We prove a priori error estimates for our approach. Numerical results are presented for linear and quadratic finite element methods.

Kim Ngan Le, William McLean, Bishnu Lamichhane

An initial-boundary value problem for the time-fractional diffusion equation is discretized in space using continuous piecewise-linear finite elements on a polygonal domain with a re-entrant corner. Known error bounds for the case of a convex polygon break down because the associated Poisson equation is no longer $H^2$-regular. In particular, the method is no longer second-order accurate if quasi-uniform triangulations are used. We prove that a suitable local mesh refinement about the re-entrant corner restores second-order convergence. In this way, we generalize known results for the classical heat equation due to Chatzipantelidis, Lazarov, Thomée and Wahlbin.