𝑝-robust equivalence of global continuous constrained and local discontinuous unconstrained approximation, a 𝑝-stable local commuting projector, and optimal elementwise ℎ𝑝 approximation estimates in H(^cp)

Let an open bounded Lipschitz polygon or polyhedron <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , a function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic v"> <mml:semantics> <mml:mi mathvariant="bold-italic">v</mml:mi> <mml:annotation encoding="application/x-tex">\boldsymbol {v}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the Sobolev space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic upper H left-parenthesis normal d normal i normal v comma normal upper Omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="bold-italic">H</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">d</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">v</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\boldsymbol {H}(\mathrm {div},\Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and a simplicial mesh of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be given. We prove the equivalence of two piecewise (Raviart–Thomas) polynomial best approximations of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic v"> <mml:semantics> <mml:mi mathvariant="bold-italic">v</mml:mi> <mml:annotation encoding="application/x-tex">\boldsymbol {v}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic upper L squared"> <mml:semantics> <mml:msup> <mml:mi mathvariant="bold-italic">L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\boldsymbol {L}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -norm: (1) globally on the whole computational domain <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega"> <mml:semantics> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:annotation encoding="application/x-tex">\Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , with the normal trace continuity requirement and a divergence constraint; (2) locally on each mesh element, without any interelement continuity requirement and without any constraint on the divergence. The former (global-best continuous constrained piecewise polynomial approximation) arises in numerical methods for partial differential equations related to the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic upper H left-parenthesis normal d normal i normal v comma normal upper Omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="bold-italic">H</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">d</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">v</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> Ω </mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\boldsymbol {H}(\mathrm {div},\Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> space, whereas the latter (local-best discontinuous unconstrained piecewise polynomial approximation) is a key quantity in approximation theory. Crucially, we establish <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -robustness in that the equivalence constant only depends on the mesh shape regularity and the spatial dimension. This improves the recent result of Ern, Gudi, Smears, and Vohralík [IMA J. Numer. Anal. 42 (2022), pp. 1023–1049], where the equivalence constant was possibly dependent on the underlying polynomial degree. Consequently, we obtain fully <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="h"> <mml:semantics> <mml:mi>h</mml:mi> <mml:annotation encoding="application/x-tex">h</mml:annotation> </mml:semantics> </mml:math> </inline-formula> - and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> - (mesh-size- and polynomial-degree-) optimal approximation estimates under the minimal Sobolev regularity only requested separately on each mesh element. These two results immediately follow by our construction of an operator from the infinite-dimensional Sobolev space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold-italic upper H left-parenthesis normal d normal i normal v comma normal upper Omega right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="bold-italic">H</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">d</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">v</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal"> Ω