A Stable Mixed Element Method for the Biharmonic Equation with First-Order Function Spaces

This paper studies the mixed element method for the boundary value problem of the biharmonic equation Δ 2 ⁢ u = f {Δ²u=f} in two dimensions. We start from a u ∼ ∇ ⁡ u ∼ ∇ 2 ⁡ u ∼ div ⁢ ∇ 2 ⁡ u {u~∇ u~∇²u∇²u} formulation that is discussed in [4] and construct its stability on H 0 1 ⁢ ( Ω ) × H ~ 0 1 ⁢ ( Ω ) × L ¯ sym 2 ⁢ ( Ω ) × H - 1 ⁢ ( div , Ω ) {H¹₀(Ω)̃¹₀(Ω)̄sym^% {2}(Ω)× H⁻¹(div,Ω)} . Then we utilise the Helmholtz decomposition of H - 1 ⁢ ( div , Ω ) {H⁻¹(div,Ω)} and construct a new formulation stable on first-order and zero-order Sobolev spaces. Finite element discretisations are then given with respect to the new formulation, and both theoretical analysis and numerical verification are given.