An hp-Adaptive Strategy Based on Locally Predicted Error Reductions

2.1 Low-Dimensional Enrichments

Consider the special case where the subspace 𝕎 ⊆ 𝕏 introduced above is spanned by finitely many basis functions ϕ 1 , … , ϕ N ∈ 𝕎 , with N ≥ 1 , i.e.

Moreover, for an index subset ℐ loc ⊂ N ¯ , we let

for instance, in the specific context of hp-discretizations to be discussed later on, the spaces 𝕎 and 𝕎 loc will take the roles of an hp-finite element space and a locally supported subspace, respectively. We can then define a linear projection operator

which enables a decomposition of the solution u 𝕎 ∈ 𝕎 of (2.1) into a low dimensional and a remaining part

respectively, with

We aim to improve the approximation u 𝕎 of the (full space) solution u 𝕏 ∈ 𝕏 of (2.3) by enriching or replacing the local space 𝕎 loc by a subspace

of dimension L + 1 , where

is a (small) set of linearly independent elements in 𝕏 . If 𝕎 loc ⊂ 𝕐 , then we call 𝕐 a local enrichment space, otherwise 𝕐 is a local replacement space; the functions ξ 1 , … , ξ L will in both cases be referred to as (local) enrichment functions.

We now introduce the low-dimensional problem

Since dim ⁡ 𝕐 = L + 1 , there is a unique ϵ ∈ ℝ and some uniquely determined y 𝝃 ∈ span ⁡ { ξ 1 , … , ξ L } such that we can express the solution u 𝕐 of (2.9) in terms of a linear combination

Then, defining the errors

we prove the following result for the predicted error reduction Δ ⁢ e 𝕎 , 𝕐 , given by

For a related result in the specific context of a multilevel solver for linear diffusion models, we mention the work [23].

The hp-adaptive approach presented in Section 3 below can be seen as a local reduction strategy. We emphasize, however, that our method, in contrast to more classical approaches, always involves (at least one) global (but computationally inexpensive) mode in order to incorporate a minimal amount of global information on the underlying problem. Similar to the present section, in order to provide maximal flexibility for the choice of local spaces and associated enrichment functions in the hp-context, the theoretical building blocks will be derived in abstract form.

2.2 Linear Algebra Representation

We will now illustrate how the difference of the residuals ρ 𝕎 ⁢ ( y 𝝃 ) and ρ 𝕐 ⁢ ( u 𝕎 loc ) from Proposition 1 can be computed by means of linear algebra. To this end, for { ξ 1 , … , ξ L } from (2.8), we introduce the matrix 𝗔 = ( A i ⁢ j ) ∈ ℝ L × L with entries

(2.13)

and the (column) vectors 𝗯 , 𝗰 ∈ ℝ L , which are given component-wise by

respectively. Furthermore, we let

2.3 Assembling Aspects

We aim to apply the abstract framework developed in the previous sections to the finite element context. For this purpose, we let 𝕏 = 𝕏 ⁢ ( Ω ) be a Hilbert function space defined over a bounded open domain Ω ⊂ ℝ d , d ≥ 1 . We present the assembling of the matrix 𝗔 and the vectors 𝗯 , 𝗰 occurring in the linear system (2.14) in the specific case where 𝕎 ⊆ 𝕏 is a subspace of finite dimension N := dim ⁡ 𝕎 with the basis { ϕ 1 , … , ϕ N } . Let 𝒟 be a decomposition of Ω into closed subsets K ⊆ Ω , i.e.

where int ⁡ ( K ) ∩ int ⁡ ( K ′ ) = ∅ for any K , K ′ ∈ 𝒟 with K ≠ K ′ . For any K ∈ 𝒟 , consider the (local) Hilbert space consisting of all restrictions of functions v ∈ 𝕏 to K, i.e. 𝕏 K := { v | K : v ∈ 𝕏 } . Moreover, let { ζ 1 K , … , ζ M K K } ⊆ 𝕏 K be a set of functions such that there exist representation matrices 𝗖 K = ( c i ⁢ j K ) ∈ ℝ N × M K and 𝗗 K = ( d i ⁢ j K ) ∈ ℝ L × M K satisfying

with { ξ 1 , … , ξ L } from (2.8). Finally, let the bilinear form a ⁢ ( ⋅ , ⋅ ) as well as the linear form b ⁢ ( ⋅ ) be decomposable in the sense that

for some (local) bilinear forms a K : 𝕏 K × 𝕏 K → ℝ and (local) linear forms b K : 𝕏 K → ℝ . For any K ∈ 𝒟 we introduce the (local) matrix 𝗔 K = ( A i ⁢ j K ) ∈ ℝ M K × M K with entries

and the (local) vector 𝗯 K = ( b i K ) ∈ ℝ M K , which is given component-wise by

Let 𝘂 ~ = ( u 1 , … , u N ) ⊤ ∈ ℝ N denote the uniquely determined coefficient vector of u ~ 𝕎 from (2.6) with respect to the basis { ϕ 1 , … , ϕ N } of 𝕎 , i.e. u ~ 𝕎 = ∑ i ∈ N ¯ u i ⁢ ϕ i . Note that u i = 0 for i ∈ ℐ loc . Then the matrix 𝗔 ∈ ℝ L × L and the vectors 𝗯 , 𝗰 ∈ ℝ L from the linear system (2.14) can be assembled by the (local) quantities 𝗔 K , 𝗖 K , 𝗗 K , and 𝗯 K as the following result shows.

3.1 Polynomial Spaces on the Reference Element

For any j ∈ ℕ 0 we introduce the 1-dimensional functions ψ j : [ - 1 , 1 ] → ℝ , given by

where L j : [ - 1 , 1 ] → ℝ denotes the j-th Legendre polynomial, normalized such that L j ⁢ ( - 1 ) = ( - 1 ) j , j ≥ 1 ; we recall the Bonnet recursion formula

as well as the relation

Figure 1

The functions ψ 0 , … , ψ 4 on [ - 1 , 1 ] .

see, e.g., [30, Section 3.1 and A.4]. The basis functions ψ 0 and ψ 1 are referred to as nodal or external shape functions whereas the high-order polynomials ψ j , for j ≥ 2 , are called internal modes (based on the fact that ψ j ⁢ ( ± 1 ) = 0 for j ≥ 2 ). We display the functions ψ 0 , … , ψ 4 in Figure 1.

Furthermore, in the multi-dimensional case, for any multi-index 𝒋 = ( j 1 , … , j d ) ∈ ℕ 0 d , we define the functions ψ ^ 𝒋 : Q ^ → ℝ by

Moreover, for any k ∈ ℕ , let ℚ k ⁢ ( Q ^ ) := span ⁡ { ψ ^ 𝒋 : 𝒋 ∈ k ¯ 0 d } signify the usual space of all polynomials up to degree k ∈ ℕ in each coordinate direction on the reference element Q ^ .

3.2 hp-Finite Element Space

The vertices of the reference element Q ^ will be labeled in terms of a multi-index notation. Specifically, to each 𝒊 ∈ { 0 , 1 } d , we associate a corresponding vertex

Similarly, we consider physical elements Q ⊂ Ω whose corner points are indexed by 𝘃 𝒊 ∈ ℝ d , i = 1 , … , 2 d ; we call Q a transformed hexahedron if the map F Q : Q ^ → Q defined by

is bijective, where we write | 𝒊 | := i 1 + ⋯ + i d to signify the order of a multi-index 𝒊 = ( i 1 , … , i d ) ∈ ℕ 0 d . Note that it holds F Q ⁢ ( 𝘃 ^ 𝒊 ) = 𝘃 𝒊 for all 𝒊 ∈ { 0 , 1 } d . The notation is illustrated in Figure 2 for the 2-dimensional case.

Figure 2

The bijective mapping F Q : Q ^ → Q for the 2-dimensional case.

For the purpose of introducing finite element subspaces 𝕎 ⊆ 𝕏 , cf. (3.1), following our abstract framework in Section 2.3, we let 𝒬 be a decomposition of Ω into transformed hexahedrons with the following additional property: If Q 1 ∩ Q 2 ≠ ∅ for Q 1 , Q 2 ∈ 𝒬 with Q 1 ≠ Q 2 , then Q 1 ∩ Q 2 represents a ( d - r ) -dimensional face of Q 1 or Q 2 for some r ∈ d ¯ ; we call any faces of dimension zero, one and two vertices, edges and faces, respectively. Furthermore, let us assume that there is no change of the type of boundary conditions within the ( d - r ) -dimensional faces of one element Q ∈ 𝒬 . We now define the hp-finite element space

associated with the decomposition 𝒬 of Ω, where p Q ≥ 1 represents an (isotropic) local polynomial degree on each element Q ∈ 𝒬 .

3.3 Constraint Coefficients

In the adaptive finite element procedure we shall represent the functions ψ ^ 𝒊 from (3.3), defined on the reference element Q ^ , in terms of functions defined on sub-hexahedra of Q ^ . These representations can be computed efficiently by means of so-called constraint coefficients, see [7, 26]. To explain this concept, consider a sub-hexahedron

where I k = [ a k , b k ] , - 1 ≤ a k < b k ≤ 1 , are 1-dimensional intervals for each k ∈ d ¯ . Then we define the functions

with the bijective element map F T : Q ^ → T , cf. (3.4). Here, due to the tensor structure (3.6) of T, we emphasize that F T is the composition of a dilation and a translation, wherefore we observe the identity

In particular, we infer that the functions ψ ~ 𝒋 in (3.7) constitute a polynomial basis on T. Therefore, there exist uniquely determined constraint coefficients b 𝒊 ⁢ 𝒋 T ∈ ℝ such that

where the sums are taken over all multi-indices 𝒋 = ( j 1 , … , j d ) ∈ ℕ 0 d with j k ≤ i k for all k ∈ d ¯ .

The ensuing result shows that the multi-dimensional constraint coefficients can be expressed in terms of tensor-products of the associated 1-dimensional quantities for which recursion formulas are stated in the Appendix.