𝐶¹ virtual element methods on polygonal meshes with curved edges

In this work we design a novel <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Superscript 1"> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>1</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">C^1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -conforming virtual element method of arbitrary order <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k greater-than-or-equal-to 2"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k \geq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , to solve the biharmonic problem on a domain with curved boundary and internal curved interfaces in two dimensions. By introducing a suitable stabilizing form, we develop a rigorous interpolation, stability and convergence analysis obtaining optimal error estimates in the energy norm. Finally, we validate the theoretical findings through numerical experiments.