Gradient‐based nodal limiters for artificial diffusion operators in finite element schemes for transport equations

Abstract

Summary This paper presents new linearity‐preserving nodal limiters for enforcing discrete maximum principles in continuous (linear or bilinear) finite element approximations to transport problems with steep fronts. In the process of algebraic flux correction, the oscillatory antidiffusive part of a high‐order base discretization is decomposed into a set of internodal fluxes and constrained to be local extremum dim inishing. The proposed nodal limiter functions are designed to be continuous and satisfy the principle of linearity preservation that implies the preservation of second‐order accuracy in smooth regions. The use of limited nodal gradients makes it possible to circumvent angle conditions and guarantee that the discrete maximum principle holds on arbitrary meshes. A numerical study is performed for linear convection and anisotropic diffusion problems on uniform and distorted meshes in two space dimensions. Copyright © 2017 John Wiley & Sons, Ltd.

Monotonicity-preserving finite element schemes based on differentiable nonlinear stabilization

“…This property is related to high-order convergence on smooth regions [25]. A modification of the nonlinear diffusion in [23] that also satisfies this property is proposed in [24]. (2) The second ingredient is the amount of diffusion to be introduced on shocks, which is the amount of diffusion introduced in a first order linear scheme.…”

On algebraically stabilized schemes for convection-diffusion-reaction problems

“…First we prove the following general form of the DMP, which generalizes a result proved in [9]. Theorem 3 Let (18), (19), and (29)- (33) hold and let Assumptions (A1) and (A2) be satisfied. Consider any nonempty set R ⊂ {1, .…”

“…, M }. If u i is a strict local extremum of U with respect to S i from ( 32), (33), i.e., Proof The proof is basically the same as in [25]. Since it is short, we repeat it for completeness.…”

“…A class of methods that has been developed intensively in recent years is the class of algebraically stabilized schemes, e.g., see [3,8,17,[28][29][30][31][32][33][34]36]. The origins of this approach can be tracked back to [10,40].…”

Some remarks concerning stabilization techniques for convection-diffusion problems

“…The basic approach is an analogy of the FCT scheme. For suitable limiter functions the AFC schemes satisfy the discrete maximum principle and linearity preservation on arbitrary meshes (see [12] and [1]). This implies the preservation of second-order accuracy in smooth regions.…”