-weak Galerkin Finite Element Method for the Biharmonic Equation

A $$C^0$$C0-weak Galerkin (WG) method is introduced and analyzed in this article for solving the biharmonic equation in 2D and 3D. A discrete weak Laplacian is defined for $$C^0$$C0 functions, which is then used to design the weak Galerkin finite element scheme. This WG finite element formulation is symmetric, positive definite and parameter free. Optimal order error estimates are established for the weak Galerkin finite element solution in both a discrete $$H^2$$H2 norm and the standard $$H^1$$H1 and $$L^2$$L2 norms with appropriate regularity assumptions. Numerical results are presented to confirm the theory. As a technical tool, a refined Scott-Zhang interpolation operator is constructed to assist the corresponding error estimates. This refined interpolation preserves the volume mass of order $$(k+1-d)$$(k+1-d) and the surface mass of order $$(k+2-d)$$(k+2-d) for the $$P_{k+2}$$Pk+2 finite element functions in $$d$$d-dimensional space.

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