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

Abstract

We present a non-conforming least squares method for approximating solutions of second order elliptic problems with discontinuous coefficients. The method is based on a general Saddle Point Least Squares (SPLS) method introduced in previous work based on conforming discrete spaces. The SPLS method has the advantage that a discrete inf − sup condition is automatically satisfied for standard choices of test and trial spaces. We explore the SPLS method for non-conforming finite element trial spaces which allow higher order approximation of the fluxes. For the proposed iterative solvers, inversion at each step requires bases only for the test spaces. We focus on using projection trial spaces with local projections that are easy to compute. The choice of the local projections for the trial space can be combined with classical gradient recovery techniques to lead to quasi-optimal approximations of the global flux. Numerical results for 2D and 3D domains are included to support the proposed method.2000 Mathematics Subject Classification. 74S05, 74B05, 65N22, 65N55.

Efficient discretization and preconditioning of the singularly perturbed Reaction Diffusion problem

“…In this paper, we assume that V 0 := Ker(B) = {0}. Most of the considerations in this section extend to a nontrivial kernel V 0 , see [9]. 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].…”

“…Let V h be a finite element subspace of V . Following [8,9], we provide a general construction of discrete trial spaces M h , defined using the operator B associated with the original problem (3.1). Let Mh ⊂ Q be a finite dimensional subspace equipped with the inner product (•, •) h .…”

“…To improve the rate of convergence (in the energy norm) for the case ε < ε 0 , we proceed by using two main tools. The first is a mixed formulation that is suitable to Saddle Point Least Squares (SPLS) discretization [7,8,9,11]. This allows for a higher order approximation of the flux ∇u using the same discretization spaces as in the standard discretization of (1.1).…”

“…Many aspects of the SPLS formulation are also common to the DPG approach [13,17,21,22,24,27]. The SPLS method as a general method for solving mixed variational formulations is presented in [7,8,9,11]. In this paper, we also combine SPLS formulation with the concept of optimal test norm [16,19,20,21,24,25,27] and a general preconditioning technique, introduced in [7], in order to improve the stability and approximability of the final discretization process.…”

Recent Advances in Least-Squares and Discontinuous Petrov–Galerkin Finite Element Methods

“…A discrete test space is paired with a discrete trial space using the operator associated with the bilinear form in such a way that this pair automatically satisfies the discrete inf-sup condition. Extension of this methodology to PDEs with discontinuous coefficients arising in elliptic interface problems, via a non-conforming trial space is performed in [1]. They show that a higher-order approximation of the fluxes is achieved.…”