A construction of canonical nonconforming finite element spaces for elliptic equations of any order in any dimension

A unified construction of canonical <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H Superscript m"> <mml:semantics> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">H^m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-nonconforming finite elements is developed for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-dimensional simplices for any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m comma n greater-than-or-equal-to 1"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m, n \geq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Consistency with the Morley-Wang-Xu elements [Math. Comp. 82 (2013), pp. 25–43] is maintained when <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m less-than-or-equal-to n"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≤</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">m \leq n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. In the general case, the degrees of freedom and the shape function space exhibit well-matched multi-layer structures that ensure their alignment. Building on the concept of the nonconforming bubble function, the unisolvence is established using an equivalent integral-type representation of the shape function space and by applying induction on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The corresponding nonconforming finite element method applies to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 m"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">2m</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-th order elliptic problems, with numerical results for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m equals 3"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m=3</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="m equals 4"> <mml:semantics> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">m=4</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in 2D supporting the theoretical analysis.