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The RT0/P1 least-squares family is conforming

Statement

Assume Countable Choice. The finite-mesh family $R T_{0} (T_{h}) \times (P_{1} (T_{h}) \cap H_{0}^{1} (Ω))$ from First-order least-squares formulation of the Dirichlet Poisson problem is a finite-dimensional subspace of $H (d i v; Ω) \times H_{0}^{1} (Ω)$ .

Facts & Assumptions

Given: A finite conforming triangular mesh.

[A1]

Countable Choice is inherited by Sobolev interfaces. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

[F1]

The RT0/P1 trial factors and boundary condition are fixed by the definition. (First-order least-squares formulation of the Dirichlet Poisson problem)

[F2]

Matching RT0 facet fluxes imply global H(div) conformity. (RT0 facet-flux unisolvence and global H(div) conformity)

[F3]

Global H1 traces agree across interior facets. (H1 traces agree across interior mesh facets)

Proof

1.1A1F1F2algebra

Local RT0 fields and their divergences are polynomials; shared normal flux degrees of freedom agree by [F2]. Their global distributional divergence is therefore the piecewise $L^{2}$ divergence, so the flux factor lies in H(div).

2.1F1F3step 1.1algebra∎

The scalar factor is globally continuous piecewise affine and has H1_0 boundary condition; [F3] gives the interior trace convention. The finite mesh has finitely many polynomial coefficients, so the product space is finite dimensional.

Depends on

Used by

The conforming RT0/P1 least-squares Poisson minimizer is unique and best approximating Theorem

Dependency tree · two levels

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Sources

Peter Bochev and Max Gunzburger, Least-squares finite element methods (standard reference, not scraped)
J. Bond, R. Chaudhry, L. Cyr, and L. Olson, A First-Order System Least-Squares Finite Element Method for the Poisson-Boltzmann Equation (standard reference, not scraped)