A Hybridizable and Superconvergent Discontinuous Galerkin Method for Biharmonic Problems

• DOI: 10.1007/s10915-009-9279-z
• Publication Date: 2009-02-23T14:47:30Z
• Authors: Bernardo Cockburn; Bo Dong; Johnny Guzmán
• Keywords: Superconvergence; Biharmonic equation; Approximations of π

In this paper, we introduce and analyze a new discontinuous Galerkin method for solving the biharmonic problem Δ2 u=f. The method has two main, distinctive features, namely, it is amenable to an efficient implementation, and it displays new superconvergence properties. Indeed, although the method uses as separate unknowns u,? u,Δu and ?Δu, the only globally coupled degrees of freedom are those of the approximations to u and Δu on the faces of the elements. This is why we say it can be efficiently implemented. We also prove that, when polynomials of degree at most k?1 are used on all the variables, approximations of optimal convergence rates are obtained for both u and ? u; the approximations to Δu and ?Δu converge with order k+1/2 and k?1/2, respectively. Moreover, both the approximation of u as well as its numerical trace superconverge in L 2-like norms, to suitably chosen projections of u with order k+2 for k?2. This allows the element-by-element construction of another approximation to u converging with order k+2 for k?2. For k=0, we show that the approximation to u converges with order one, up to a logarithmic factor. Numerical experiments are provided which confirm the sharpness of our theoretical estimates.