A locally conservative least‐squares method for Darcy flows

Abstract

SUMMARYLeast-squares finite-element methods for Darcy flow offer several advantages relative to the mixedGalerkin method: the avoidance of stability conditions between finite-element spaces, the efficiency of solving symmetric and positive definite systems, and the convenience of using standard, continuous nodal elements for all variables. However, conventional C 0 implementations conserve mass only approximately and for this reason they have found limited acceptance in applications where locally conservative velocity fields are of primary interest. In this paper, we show that a properly formulated compatible least-squares method offers the same level of local conservation as a mixed method. The price paid for gaining favourable conservation properties is that one has to give up what is arguably the least important advantage attributed to least-squares finite-element methods: one can no longer use continuous nodal elements for all variables. As an added benefit, compatible least-squares methods inherit the best computational properties of both Galerkin and mixed-Galerkin methods and, in some cases, yield identical results, while offering the advantages of not having to deal with stability conditions and yielding positive definite discrete problems. Numerical results that illustrate our findings are provided.

Split least-squares finite element methods for linear and nonlinear parabolic problems

“…The key point used to explain the split of the procedure is Lemma 2.1 which was obtained by integration by parts. Similar results have been found and used by [8] to prove the coercivity of least-squares bilinear formats and by [2,3] to establish connections between least-squares and mixed methods. The last two papers also show that not only is the pressure the same as in the Galerkin method, but also the flux is the same as in the mixed method under some conditions on the finite element spaces.…”

“…Results similar to Lemma 2.1 or Lemma 2.2 have been found and used by [8] to prove the coercivity of leastsquares bilinear formats and by [2,3] to establish connections between least-squares and mixed methods.…”

Principles of Mimetic Discretizations of Differential Operators

“…It also highlighted the role of differential forms and algebraic topology in the design and analysis of compatible discretizations. The recent work in [2,8,9,10,22,29,30,39,44,47,52,53,58] and the papers in this volume further affirm that these tools are gaining wider acceptance among mathematicians and engineers. For instance, FE methods that have traditionally relied upon nonconstructive variational [6,18] stability criteria 1 now are being derived by topological approaches that reveal physically relevant degrees of freedom and their proper encoding.…”

“…For further details on mimetic discretizations with weak constitutive laws and their connection to least-squares minimization principles we refer to [7,8,9]. Examples of this idea in magnetostatics can be found in [14] and [20].…”

A two-level enriched finite element method for a mixed problem

“…This leads to some very popular choices, such as equal order interpolation spaces and the simplest element P 1 /P 0 to be out of reach, or it prevents nodal values to be chosen as degrees of freedom if some physical properties (such as local conservation of mass) are to be satisfied by the numerical method (cf. [9]). …”