Two‐Grid hp‐DGFEMs on Agglomerated Coarse Meshes

We generalise the a priori error analysis of two‐grid hp‐version discontinuous Galerkin finite element methods for strongly monotone second‐order quasilinear elliptic partial differential equations to the case when coarse meshes consisting of general agglomerated polytopic elements are employed.

The two-grid method was originally introduced by Xu [1,2]. The key idea of this approach, in the context of numerically approximating nonlinear partial differential equations (PDEs), is to first compute a numerical approximation of the nonlinear PDE on a coarse mesh/approximation space, and subsequently employ this solution to linearize the underlying problem on the fine mesh/approximation space; in this way only a linear solve is required on the fine mesh/approximation space. In the context of hp-version DGFEMs, in [3] and [4] we have considered the application of the two-grid approach to both scalar strongly monotone second-order quasilinear PDEs of the form (1) and non-Newtonian fluids, respectively; in both cases the coarse and fine spaces employ standard meshes employing simplices/tensor-product elements. In this article, we generalize this to the case when general polytopic coarse elements, generated by agglomerating fine mesh elements, are employed.

Two-grid hp-version IIP DGFEM
We write T h = {κ} to denote the fine mesh consisting of simplices/tensor-product elements of local mesh size h κ = diam(κ), κ ∈ T h . Similarly, T H = {K} denotes the coarse mesh consisting of polytopic elements K constructed by agglomerating elements κ ∈ T h ; H K = diam(K), K ∈ T H . We assume that T h is of bounded local variation. Writing p = {p κ : κ ∈ T h } and P = {P K : K ∈ T H } to denote the polynomial orders defined over T h and T H , respectively, (p is assumed to be of bounded local variation) we write V hp = {v ∈ L 2 (Ω) : v| κ ∈ P pκ (κ), κ ∈ T h } and V HP = {v ∈ L 2 (Ω) : v| K ∈ P P K (K), K ∈ T H }, where P p (κ) denotes the space of all polynomials of total degree p on κ.
We write F h and F H to denote the set of all faces in the meshes T h and T H , respectively. Furthermore, we write { {·} } and [[·]] to denote suitable average and jump operators, respectively, which are defined on either F h or F H ; see [3] for details. With this notation, we first introduce the following standard IIP DGFEM on the fine mesh T h , for the numerical approximation of the problem (1) and ∇ h is used to denote the broken gradient operator, defined elementwise. Given a face polynomial degree function p F and a face mesh size function h F , F ∈ F h , the interior penalty parameter σ hp is given by where γ hp > 0 is a sufficiently large constant, cf. [3]. The two-grid IIP DGFEM is given by: Here, A HP (u; u, v) is defined analogously to A hp (u; u, v), but with a modified interior penalty parameter σ HP , cf. [5].
Section 18: Numerical methods of differential equations

Error analysis
For the proceeding error analysis, we require the following definitions and assumptions, cf. [5]. Definition 3.1 For K ∈ T H we write F K to be the set of all possible d-simplices contained in K and having at least one face in common with K; we write K F to denote a simplex belonging to F K which shares with K ∈ T H the face F ⊂ ∂K.
Assumption 3.2 For any K ∈ T H , there exists a set of non-overlapping d-dimensional simplices {K F } ⊂ F K contained within K, such that for all F ⊂ ∂K, the condition H K ≤ C s d|K F ||F | −1 holds, where C s is a positive constant, which is independent of the discretization parameters, the number of faces that the element possesses, and the measure of F . If the analytical solution u ∈ H 1 (Ω) to (1) satisfies u| κ ∈ H lκ (κ), l κ ≥ 2, and u| K ∈ H L K (K), L K ≥ 3 /2, for K ∈ T H , such that Eu| K ∈ H L K (K), where K ∈ T H with K ⊂ K; then, writing v 2 where G K (H K , P K ) := (P K + P 2 K )H −1 K max F ⊂∂κ σ −1 HP | F + H K P −1 K max F ⊂∂K σ HP | F , S K = min(P K + 1, L K ), for K ∈ T H , s κ = min(p κ + 1, l κ ), for κ ∈ T h , and C is a positive constant independent of u, h, H, p, and P , but depends on the constants m µ , M µ from the monotonicity properties of µ(·). Finally, E denotes the extension operator defined in [7].