A HYBRIDIZABLE DISCONTINUOUS GALERKIN METHOD FOR THE p-LAPLACIAN

作者：Cockburn Bernardo^*; Shen Jiguang^*

来源：SIAM Journal on Scientific Computing, 2016, 38(1): A545-A566.

DOI：10.1137/15M1008014

• hybridization
• discontinuous Galerkin methods
• degenerate elliptic equations
• p-aplacian operator
• nonlinear minimization algorithms
• preconditioning
• postprocessing

We propose the first hybridizable discontinuous Galerkin method for the p-Laplacian equation. When using polynomials of degree k >= 0 for the approximation spaces of u, del u, and vertical bar del u vertical bar(p-2)del u, the method exhibits optimal k + 1 order of convergence for all variables in L-1- and L-p-norms in our numerical experiments. For k >= 1, an elementwise computation allows us to obtain a new approximation u(h)* that converges to u with order k + 2. We rewrite the scheme as discrete minimization problems in order to solve them with nonlinear minimization algorithms. The unknown of the first problem is the approximation of u on the skeleton of the mesh but requires solving nonlinear local problems. The second problem has the approximation on the elements as an additional unknown but it only requires solving linear local problems. We present numerical results displaying the convergence properties of the methods, demonstrating the utility of using frozencoefficient preconditioners, and indicating that the second method is superior to the first one even though it has more unknowns.

• 出版日期2016