Analysis of the boundary conditions for the ultraweak-local discontinuous Galerkin method of time-dependent linear fourth-order problems

Abstract

In this paper, we study the ultraweak-local discontinuous Galerkin (UWLDG) method for time-dependent linear fourth-order problems with four types of boundary conditions. In one dimension and two dimensions, stability and optimal error estimates of order k + 1 k+1 are derived for the UWLDG scheme with polynomials of degree at most k k ( k ≥ 1 k\ge 1 ) for solving initial-boundary value problems. The main difficulties are the design of suitable penalty terms at the boundary for numerical fluxes and the construction of projections. More precisely, in two dimensions with the Dirichlet boundary condition, an elaborate projection of the exact boundary condition is proposed as the boundary flux, which, in combination with some proper penalty terms, leads to the stability and optimal error estimates. For other three types of boundary conditions, optimal error estimates can also be proved for fluxes without any penalty terms when special projections are designed to match different boundary conditions. Numerical experiments are presented to confirm the sharpness of theoretical results.