A finite element elasticity complex in three dimensions
Long Chen, Xuehai Huang
Abstract
A finite element elasticity complex on tetrahedral meshes and the corresponding commutative diagram are devised. The <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">H^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> conforming finite element is the finite element developed by Neilan for the velocity field in a discrete Stokes complex. The symmetric div-conforming finite element is the Hu-Zhang element for stress tensors. The construction of an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H left-parenthesis i n c right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>inc</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H(\operatorname {inc})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -conforming finite element of minimum polynomial degree <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="6"> <mml:semantics> <mml:mn>6</mml:mn> <mml:annotation encoding="application/x-tex">6</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for symmetric tensors is the focus of this paper. Our construction appears to be the first <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H left-parenthesis i n c right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>inc</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H(\operatorname {inc})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -conforming finite elements on tetrahedral meshes without further splitting. The key tools of the construction are the decomposition of polynomial tensor spaces and the characterization of the trace of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i n c"> <mml:semantics> <mml:mi>inc</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {inc}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> operator. The polynomial elasticity complex and Koszul elasticity complex are created to derive the decomposition. The trace of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i n c"> <mml:semantics> <mml:mi>inc</mml:mi> <mml:annotation encoding="application/x-tex">\operatorname {inc}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> operator is induced from a Green’s identity. Trace complexes and bubble complexes are also derived to facilitate the construction. Two-dimensional smooth finite element Hessian complex and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d i v d i v"> <mml:semantics> <mml:mrow> <mml:mi>div</mml:mi> <mml:mo> </mml:mo> <mml:mi>div</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {div}\operatorname {div}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> complex are constructed.