A saddle point least squares approach for primal mixed formulations of second order PDEs

Abstract

We present a Saddle Point Least Squares (SPLS) method for discretizing second order elliptic problems written as primal mixed variational formulations. A stability LBB condition and a data compatibility condition at the continuous level are automatically satisfied. The proposed discretization method follows a general SPLS approach and has the advantage that a discrete inf − sup condition is automatically satisfied for standard choices of the test and trial spaces. For the proposed iterative processes a nodal basis for the trial space is not required. Efficient preconditioning techniques that involve inversion only on the test space can be considered. Stability and approximation properties for two choices of discrete spaces are investigated. Applications of the new approach include discretization of second order problems with highly oscillatory coefficient, interface problems, and higher order approximation of the flux for elliptic problems with smooth coefficients.2000 Mathematics Subject Classification. 74S05, 74B05, 65N22, 65N55.and that the following inf − sup condition holds,With the form b, we associate the linear operators B : V → Q * and B * : Q → V * defined byIt is known that under the assumption (2.3), the operator C −1 B : V → Q is onto, see e.g., [3]. We let V 0 be the kernel of B or C −1 B, i.e., V 0 = Ker(B) = {v ∈ V | Bv = 0} = {v ∈ V | C −1 Bv = 0}.

Saddle point least squares preconditioning of mixed methods

“…The benefit of using the projection type trial space is that it could lead to a better approximation of the continuous solution p. Indeed, for the case when preconditioning is not used, super-convergence of p − p h is observed, see [4,6,7]. Using uniform preconditioners and Theorems 3.3 and 3.6, we expect the same order of super-convergence for p − p h .…”

A Non-conforming Saddle Point Least Squares Approach for an Elliptic Interface Problem

“…We take V h ⊂ V = H 1 0 (Ω) to be the space of continuous piecewise polynomials of degree k with respect to the interface-fitted triangular mesh T h . We note that while the no projection trial space case is similar with the work presented in [11], the projection trial space is analyzed using the nonconforming trial space setting and leads to new stability and approximability estimates for the discontinuous coefficients (or interface) case.…”

“…which implies (2.4) is trivially satisfied. We note that, as presented in [11], the continuity constant satisfies…”

“…The SPLS method can be also be combined with multilevel preconditioning techniques in order to address particular challenges of the PDE to be solved due to discontinuous coefficients or multidimensional domains [6]. In contrast with the SPLS work presented in [10,11], where both the test and trial spaces were chosen to be conforming finite element spaces, this paper considers trial spaces which are non-conforming finite element spaces. This allows efficient treatment of PDEs with discontinuous coefficients.…”

Efficient discretization and preconditioning of the singularly perturbed Reaction Diffusion problem

“…It is well known that if a bounded form b : V × Q → R satisfies (3.2), then problem (3.1) has a unique solution, see e.g., [3,4]. The standard saddle point reformulation of (3.1) (see [10,11,12,20]) is:…”