A Finite Element Method for the Surface Biharmonic Problem without Stabilization or Piola Transforms
For the biharmonic problem on closed smooth surfaces, we construct a nonconforming finite element method that requires neither stabilization nor the Piola transform. The reduced enriched quartic (REQ) element relies solely on vertex values and scalar edge moments. The same local construction applies to every triangle of a piecewise planar surface. The bilinear form consists of elementwise Laplacian inner products and permits standard finite element assembly. By separating the planar vertex residual from the angle defect contribution, we establish weak discrete Miranda--Talenti-type estimates on both planar meshes and closed piecewise planar surfaces. These estimates guarantee uniform stability on sufficiently refined interpolating surface meshes. We prove the well-posedness of the method, first-order convergence in the energy norm, and second-order convergence in the broken $H^1$ and $L^2$ norms. Numerical tests on three surfaces show the predicted convergence rates and examine the discrete Miranda--Talenti constants and angle defect scaling.