By Pavel B. Bochev

When you consider that their emergence, finite aspect tools have taken a spot as essentially the most flexible and strong methodologies for the approximate numerical answer of Partial Differential Equations. those equipment are utilized in incompressible fluid move, warmth, move, and different difficulties. This booklet presents researchers and practitioners with a concise consultant to the speculation and perform of least-square finite aspect tools, their strengths and weaknesses, confirmed successes, and open difficulties.

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**Example text**

Here, we again note that defining finite element subspaces that satisfy this condition is no easy matter. Null space method. 3. 26) is now given by: seek v ∈ Z such that ∇v : ∇r dΩ = f · r dΩ ∀r ∈ Z . 58). However, the conformity requirement Z h ⊂ Z implies that a function belonging to the finite element space Z h must be (at least weakly) divergence free; in practice, such spaces are difficult to construct; see [191]. Equivalent PDEs. 62) v =0 on ∂ Ω . 62). 61) is formally equivalent to the problem −∆ ΠZ v = ΠZ f, where ΠZ v denotes the projection of v onto the space of (weakly) solenoidal functions.

9) is well posed. 3. Here, we state it as a separate result because of its historical importance. 17 We again visit the finite-dimensional setting. Let U = W = W ∗ = RN , Q ∈ RN×N , f ∈ RN , Q(u; w) = wT Qu, and F(w) = wT f . A simple argument shows that the strong coercivity of Q(·, ·) is equivalent to Q being a real matrix with a positive definite symmetric part. A real, positive definite matrix is always invertible. This is the algebraic version of the statement that strong coercivity implies weak coercivity.

2) is square and nonsingular. 36) implies that, in addition to these properties, strongly coercive variational formulations give rise to positive definite linear systems. Furthermore, in the Rayleigh–Ritz setting, these systems are also symmetric. 9 points out an important difference between the conditions that guarantee the stability of discretizations of weakly and strongly coercive variational problems. In both cases, the continuity property of the bilinear form Q(·, ·) with respect to conforming discrete subspaces is inherited from the continuity property that holds on the parent spaces.