Media type: E-Article Title: Sharp anisotropic interpolation error estimates for rectangular Raviart-Thomas elements Contributor: Stynes, Martin Published: American Mathematical Society (AMS), 2014 Published in: Mathematics of Computation, 83 (2014) 290, Seite 2675-2689 Language: English DOI: 10.1090/s0025-5718-2014-02826-3 ISSN: 0025-5718; 1088-6842 Keywords: Applied Mathematics ; Computational Mathematics ; Algebra and Number Theory Origination: Footnote: Description: <p>Anisotropic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript p"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">L_p</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-norm error estimates are derived for the standard rectangular Raviart-Thomas elements <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R upper T Subscript left-bracket k right-bracket Baseline left-parenthesis upper K overTilde right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">[</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>K</mml:mi> <mml:mo stretchy="false">~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">RT_{[k]}(\tilde K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript d"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">\mathbb {R}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p element-of left-bracket 1 comma normal infinity right-bracket comma k greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">∞<!-- ∞ --></mml:mi> <mml:mo stretchy="false">]</mml:mo> <mml:mo>,</mml:mo> <mml:mtext> </mml:mtext> <mml:mi>k</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p\in [1, \infty ],\ k\ge 0</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="d greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d \ge 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Here <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K overTilde"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>K</mml:mi> <mml:mo stretchy="false">~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\tilde K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an affine image of an axi-parallel parallelotope <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The proofs are based on a variant of the classical Poincaré inequality. The estimates derived make full use of the asymmetric nature of the vector space components of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R upper T Subscript left-bracket k right-bracket Baseline left-parenthesis upper K overTilde right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>R</mml:mi> <mml:msub> <mml:mi>T</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">[</mml:mo> <mml:mi>k</mml:mi> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi>K</mml:mi> <mml:mo stretchy="false">~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">RT_{[k]}(\tilde K)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; a Shishkin mesh example demonstrates their superiority over previous estimates.</p> Access State: Open Access