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.
Bibliographical noteFunding Information:
The first author was supported in part by the National Science Foundation (Grant DMS-0411254) and by the University of Minnesota Supercomputing Institute. The third author was supported by an NSF Mathematical Science Postdoctoral Research Fellowship (DMS-0503050).
- Biharmonic problems
- Discontinuous Galerkin methods