The key highlights and insights of this content are:
The main challenge in discretizing fourth-order PDEs on surfaces is the general C0,1 continuity of the approximate surface, which makes H1 elements unsuitable for direct discretization of fourth-order differential operators.
The authors propose a continuous linear finite element method that employs a strategic utilization of a surface gradient recovery operator to compute the second-order surface derivative of a piecewise continuous linear function defined on the approximate surface.
The authors establish the stability of the proposed formulation by incorporating appropriate stabilizations, and provide optimal error estimates in both the energy norm and L2 norm despite the presence of geometric error.
The authors highlight the key difficulties in this extension, including the need to handle the discontinuity of the conormal vector across element edges and the challenges in obtaining optimal error estimates due to the violation of Galerkin orthogonality.
The authors draw inspiration from various techniques, such as the non-standard geometric error estimate (Phn lemma) and the methodology for analyzing the consistency error between the continuous and discrete bilinear forms, to overcome these challenges and establish the desired error estimates.
Numerical experiments are provided to support the theoretical results.
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