Order Two Superconvergence of the CDG Finite Elements on Triangular and Tetrahedral Meshes

Abstract

It is known that discontinuous finite element methods use more unknown variables but have the same convergence rate comparing to their continuous counterpart. In this paper, a novel conforming discontinuous Galerkin (CDG) finite element method is introduced for Poisson equation using discontinuous P k elements on triangular and tetrahedral meshes. Our new CDG method maximizes the potential of discontinuous P k element in order to improve the convergence rate. Superconvergence of order two for the CDG finite element solution is proved in an energy norm and in the L 2 norm. A local post-process is defined which lifts a P k CDG solution to a discontinuous P k+2 solution. It is proved that the lifted P k+2 solution converges at the optimal order. The numerical tests confirm the theoretic findings. Numerical comparison is provided in 2D and 3D, showing the P k CDG finite element is as good as the P k+2 continuous Galerkin finite element.

A superconvergent CDG finite element for the Poisson equation on polytopal meshes

“…Here four aligned squares may rotate together. [18] proves that (3.8) defines an 𝐸 𝑒 𝑣. [18] shows also that it preserves 𝑃 𝑘+2 (𝑈 𝑒 ) polynomials in the sense 𝐸 𝑒 𝑣 = 𝑤 if 𝑣| 𝑆 𝑖 = Π 𝑘,𝑆 𝑒 𝑤 for all 𝑤 ∈ 𝑃 𝑘+2 (𝑈 𝑒 ).…”

“…[18] proves that (3.8) defines an 𝐸 𝑒 𝑣. [18] shows also that it preserves 𝑃 𝑘+2 (𝑈 𝑒 ) polynomials in the sense 𝐸 𝑒 𝑣 = 𝑤 if 𝑣| 𝑆 𝑖 = Π 𝑘,𝑆 𝑒 𝑤 for all 𝑤 ∈ 𝑃 𝑘+2 (𝑈 𝑒 ). In 3D, the set {𝑆 𝑖 } in (3.8) contains eight aligned cubes, two in each direction.…”

“…Lemma 3.1 [18]. For 𝑘 ≥ 1, the lifting operator 𝐸 𝑒 ∶ 𝑉 ℎ → 𝑃 𝑘+2 (𝑈 𝑒 ), defined in (3.8), has an order 𝑘 + 2 accuracy, that is, for any 𝑢 ∈ 𝐻 𝑘+3 (Ω),…”

“…[2]. But a CDG method is introduced in [4, [10][11][12][13][15][16][17][18] which keeps the weak form (1.3) of the conforming finite element method, unlike the other DG methods. This paper extends the two-order superconvergent CDG method of [18], which is on triangular and tetrahedral meshes only, to general polytopal meshes in 2D and 3D.…”

“…But a CDG method is introduced in [4, [10][11][12][13][15][16][17][18] which keeps the weak form (1.3) of the conforming finite element method, unlike the other DG methods. This paper extends the two-order superconvergent CDG method of [18], which is on triangular and tetrahedral meshes only, to general polytopal meshes in 2D and 3D. Unlike other DG methods, the inter-element trace of discontinuous F I G U R E 1 A polygon is subdivided in to triangles by connecting ONLY one vertex with other vertices.…”