A Superconvergent Local Discontinuous Galerkin Method for Elliptic Problems

Abstract

In this manuscript we investigate the convergence properties of a minimal dissipation local discontinuous Galerkin(md-LDG) method for two-dimensional diffusion problems on Cartesian meshes. Numerical computations show O(h p+1 ) L 2 convergence rates for the solution and its gradient and O(h p+2 ) superconvergent solutions at Radau points on enriched p-degree polynomial spaces. More precisely, a local error analysis reveals that the leading term of the LDG error for a p-degree discontinuous finite element solution is spanned by two (p + 1)-degree right Radau polynomials in the x and y directions. Thus, LDG solutions are superconvergent at right Radau points obtained as a tensor product of the shifted roots of the (p + 1)-degree right Radau polynomial. For tensor product polynomial spaces, the first component of the solution's gradient is O(h p+2 ) superconvergent at tensor product of the roots of left Radau polynomial in x and right Radau polynomial in y while the second component is O(h p+2 ) superconvergent at the tensor product of the roots of the right Radau polynomial in x and left Radau polynomial in y. Several numerical simulations are performed to validate the theory.

Superconvergence of the local discontinuous galerkin method applied to the one‐dimensional second‐order wave equation

“…They proved that, for smooth solutions, these a posteriori error estimates at a fixed time t converge to the true spatial error in the L 2 ‐norm under mesh refinement. More recently, Adjerid and Baccouch showed that LDG solutions are superconvergent at Radau points for 2D convection‐diffusion problems. They used these results to construct asymptotically correct a posteriori error estimates.…”

A superconvergent local discontinuous Galerkin method for the second-order wave equation on Cartesian grids

“…We note that the stabilization parameter associated with the numerical trace of q is identically equal to zero in the interior of the domain. In [26], we used the same numerical fluxes to study the superconvergence properties of the LDG method for two-dimensional diffusion problems on Cartesian meshes.…”

“…For an introduction to the subject of a posteriori error estimation see the monograph of Ainsworth and Oden [19]. Superconvergence properties for finite element and DG methods have been studied in [3,5,20,2] for ordinary differential equations, [21,22,3,23] for hyperbolic problems and [24][25][26]23,[27][28][29][30][31] for diffusion and convection-diffusion problems. Several a posteriori DG error estimates are known for hyperbolic [32][33][34] and diffusive [35,36] problems.…”

“…They further used the superconvergence results to construct asymptotically correct a posteriori estimates of spatial discretization errors. More recently, Adjerid and Baccouch [26,57] showed that LDG solutions are superconvergent at Radau points for two-dimensional convection-diffusion problems. They used these results to construct asymptotically correct a posteriori error estimates.…”

Superconvergence of a full‐discrete combined mixed finite element and discontinuous Galerkin method for a compressible miscible displacement problem

“…Superconvergence is the essential character of the finite element method. Some studies of superconvergence can be founded in [6,7,[11][12][13][14][15][16][17][18][19][20]. Superconvergence of a semidiscrete combined MFE/DG approximation is investigated in [21,22].…”