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The biharmonic equation with Dirichlet and Neumann boundary conditionsdiscretized using the mixed finite element method and piecewise linear (withthe possible exception of boundary triangles) finite elements on triangularelements has been well-studied for domains in R2. Here we study the analogousproblem on polyhedral surfaces. In particular, we provide a convergence proofof discrete solutions to the corresponding smooth solution of the biharmonicequation. We obtain convergence rates that are identical to the ones known forthe planar setting. Our proof focuses on three different problems: solving thebiharmonic equation on the surface, solving the biharmonic equation in adiscrete space in the metric of the surface, and solving the biharmonicequation in a discrete space in the metric of the polyhedral approximation ofthe surface. We employ inverse discrete Laplacians to bound the error betweenthe solutions of the two discrete problems, and generalize a flat strategy tobound the remaining error between the discrete solutions and the exact solutionon the curved surface.

A mixed finite element method with piecewise linear elements for the biharmonic equation on surfaces