Convergence of nonconforming finite element approximations to first-order linear hyperbolic equations

Abstract

Finite element approximations of the first-order hyperbolic equa- tion UVu + au = / are considered on curved domains £2 C K2 . When part of the boundary of I2 is characteristic, the boundary of numerical domain, I2A , may become either an inflow or outflow boundary, so it is necessary to select an algorithm that will accommodate this ambiguity. This problem was motivated by a problem in acoustics, where an equation similar to the one above is coupled to three elliptic equations. In the last sec- tion, the acoustics problem is briefly recalled and our results for the first-order equation are used to demonstrate convergence of finite element approximations of the acoustics problem. In this paper we address some technical issues associated with approximating a first-order spatially hyperbolic equation. This problem was motivated by a problem in linear acoustics which gave rise to a system of coupled equations, one of whose principal parts was first-order in space. The acoustics problem was naturally posed in a domain, Q c R2, where a fluid entered one part of the boundary (the inflow) and exited through another portion of the boundary (the outflow). The remainder of the boundary was tangential to the mean flow, and the streamlines of this mean flow were the characteristics of a first-order equation of the form UVu + au = f, where / represents the coupling terms and U is a known (mean flow) velocity field. The numerical approximation of the solution of such problems requires tri- angulation of the domain I2, the union of the triangles giving a domain QA which approximates Q. One problem encountered with this approach is that portions of the boundary of Ci may no longer be characteristic where the cor- responding portions of Q are. This may result in triangles having one side which contains both inflow and outflow regions of the mean flow, and if the