A dual finite element complex on the barycentric refinement

2007/05/03 by Annalisa Buffa, Snorre Christiansen, Snorre H. Christiansen · 2 citations
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic Geometry and Number Theory #Combinatorics #Computer science #Curl (programming language) #Degenerate energy levels #Dual space #Duality (order theory) #Finite element method #Mathematics #Physics #Polynomial and algebraic computation #Pure mathematics #Realization (probability) #Simplicial complex #Vector space

paper · pdf · doi:10.1090/s0025-5718-07-01965-5

openalex publication_date 2007/05/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/26

Abstract

Given a two dimensional oriented surface equipped with a simplicial mesh, the standard lowest order finite element spaces provide a complex <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X Superscript bullet"> <mml:semantics> <mml:msup> <mml:mi>X</mml:mi> <mml:mo> ∙ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">X^\bullet</mml:annotation> </mml:semantics> </mml:math> </inline-formula> centered on Raviart-Thomas divergence conforming vector fields. It can be seen as a realization of the simplicial <italic>cochain</italic> complex. We construct a new complex <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y Superscript bullet"> <mml:semantics> <mml:msup> <mml:mi>Y</mml:mi> <mml:mo> ∙ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">Y^\bullet</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of finite element spaces on the barycentric refinement of the mesh which can be seen as a realization of the simplicial <italic>chain</italic> complex on the original (unrefined) mesh, such that the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper L squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathrm L²</mml:annotation> </mml:semantics> </mml:math> </inline-formula> duality is non-degenerate on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y Superscript i Baseline times upper X Superscript 2 minus i"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>Y</mml:mi> <mml:mi>i</mml:mi> </mml:msup> <mml:mo> × </mml:mo> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mo> − </mml:mo> <mml:mi>i</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">Yⁱ × X^2-i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i element-of StartSet 0 comma 1 comma 2 EndSet"> <mml:semantics> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo fence="false" stretchy="false"></mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo fence="false" stretchy="false"></mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">i∈ \0,1,2\</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In particular <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Y Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>Y</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">Y¹</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a space of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal c normal u normal r normal l"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">c</mml:mi> <mml:mi mathvariant="normal">u</mml:mi> <mml:mi mathvariant="normal">r</mml:mi> <mml:mi mathvariant="normal">l</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm curl</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -conforming vector fields which is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper L squared"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathrm L²</mml:annotation> </mml:semantics> </mml:math> </inline-formula> dual to Raviart-Thomas <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d i v"> <mml:semantics> <mml:mi>div</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname div</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -conforming elements. When interpreted in terms of differential forms, these two complexes provide a finite-dimensional analogue of Hodge duality.

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