A conforming discontinuous Galerkin finite element method for the Stokes problem on polytopal meshes

Abstract

A new discontinuous Galerkin finite element method for the Stokes equations is developed in the primary velocity‐pressure formulation. This method employs discontinuous polynomials for both velocity and pressure on general polygonal/polyhedral meshes. Most finite element methods with discontinuous approximation have one or more stabilizing terms for velocity and for pressure to guarantee stability and convergence. This new finite element method has the standard conforming finite element formulation, without any velocity or pressure stabilizers. Optimal‐order error estimates are established for the corresponding numerical approximation in various norms. The numerical examples are tested for low and high order elements up to the degree four in 2D and 3D spaces.

A Stabilizer-Free Weak Galerkin Finite Element Method for the Stokes Equations

“…Let n be the number of the edges/faces on a polygon/polyhadron. We can find the complete proof in [25] that there exists τ 0 ∈ [P j (T)] d×d , j = n+k−1, such that…”

A stabilizer free weak Galerkin finite element method on polytopal mesh: Part III

“…The idea of removing stabilizers for the WG methods in [9,11] is how to approximate weak gradient ∇ w . A polynomial of degree j is used in [9,10] to approximate weak gradient ∇ w . Here j = k + n − 1 and n is the number of sides of polytopal element.…”

A stabilizer free WG method for the Stokes equations with order two superconvergence on polytopal mesh

“…Development of stabilizer free discontinuous finite element method is desirable since it simplifies finite element formulation and reduces programming complexity. The stabilizer free WG method and the stabilizer DG method on polytopal mesh were first introduced in [11,12] for second order elliptic problems. The main idea in [11,12] is to raise the degree of polynomials used to compute weak gradient ∇ w .…”

“…The stabilizer free WG method and the stabilizer DG method on polytopal mesh were first introduced in [11,12] for second order elliptic problems. The main idea in [11,12] is to raise the degree of polynomials used to compute weak gradient ∇ w . In [11,12], gradient is approximated by a polynomial of order j = k + n − 1 where n is the number of sides of polygonal element.…”

“…The main idea in [11,12] is to raise the degree of polynomials used to compute weak gradient ∇ w . In [11,12], gradient is approximated by a polynomial of order j = k + n − 1 where n is the number of sides of polygonal element. This result has been improved in [1,2] by reducing the degree of polynomial j.…”