A Galerkin procedure for diffusion equations with boundary integral conditions

Abstract

In this paper we shall study Galerkin approximations to the solution of heat equation u t = Δu + ƒ(x, t) in Q T = Ω × (0, T], t > 0 subject to the following integral boundary condition ∂u ∂v = ∫ 0 1 K(x, t, τ)u(x, τ) dr on S T = ∂Ω × [0, T] , and the initial condition u( x, 0) = u 0( x), x ϵ Ω, where Ω ⊂R d (d ≥ 1) and ∂Ω is a smooth boundary, v( x) = ( v 1( x) 2, …, v d ( x)) is the outer-normal direction on ∂Ω, and ƒ, u 0 and K are known functions. Optimal L 2 error estimates are demonstrated for the continuous and discrete Galerkin approximations.