Hybridizable symmetric stress elements on the barycentric refinement in arbitrary dimensions

Hybridizable <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H left-parenthesis d i v right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H(div)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -conforming finite elements for symmetric tensors on simplices with barycentric refinement are developed in this work for arbitrary dimensions and any polynomial order. By employing barycentric refinement and an intrinsic tangential-normal decomposition, novel basis functions are constructed to redistribute degrees of freedom while preserving <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H left-parenthesis d i v right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">H(div)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -conformity and symmetry, and ensuring inf-sup stability. These hybridizable elements enhance computational flexibility and efficiency, with applications to mixed finite element methods for linear elasticity.