FIRST-ORDER SYSTEM LEAST-SQUARES METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS BYEONG CHUN SHIN

Let Ω be the square (−1, 1)d with d = 2 or 3. We consider the second-order elliptic boundary value problem: (1.1) − ∇ ·A∇p+ b · ∇p+ c0 p = f, in Ω, p = 0, on ΓD, n ·A∇p = 0, on ΓN, where ∂Ω = ΓD ∪ΓN denotes the boundary of Ω, A is a 2 × 2 symmetric matrix of bounded functions, f is a given continuous function, b is a bounded vector function and c0 is a given bounded function, and n is the outward unit vector normal to the boundary. We assume that the matrix A is uniformly elliptic such as (1.2) 0 < λ ξtξ ≤ ξtA(x, y)ξ ≤ Λ ξtξ <∞ for all ξ ∈ <2 and almost all (x, y) ∈ Ω̄. Setting the flux variable u = ∇p, we have the first-order system of linear equa-tions equivalent to (1.1): (1.3) − ∇ · u+ b · u+ c0 p = f, in Ω, u−A∇p = 0, in Ω, ∇×A−1u = 0, in Ω, p = 0, on ΓD, n · u = 0, on ΓN n×A−1u = 0, on ΓD. In recent years there has been lots of interest in the use of first-order system least-squares method (FOSLS) for numerical approximations of elliptic partial differen-tial equations, Stokes equations, elasticity and Navier-Stokes equations. In this paper we will provide a brief review of FOSLS around elliptic problems and Stokes equations. FOSLS is to find a minimization solution which minimizes least-squares func-tional defined by summing appropriate norms of residual equations: G(v, q; f) = ‖f + ∇ · v − b · v − c0 q‖2α + ‖v −A∇q‖2β + ‖ ∇ ×A−1v‖2