Consolidación

For the standard IP DGFEM discretisation of the quasilinear problem (3.1)–(3.2), the mesh may be automatically constructed using the hp-adaptive refinement algorithm outlined in Houston et al. [125]. In that setting, the local error indicators are defined in

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an analogous way to ηκgiven in (3.20), with uH,K and u2Gboth replaced by uh,k. In the context of the two-grid IP DGFEM discretisation defined by (3.10)–(3.11), it is necessary to refine both the fine and coarse meshes, together with their corresponding polynomial degree vectors, in order to decrease the error between u and u2G with respect to the energy normk · kh,k.

To this end, we first note that, from Theorem 3.4, we have for each fine element κ _{∈ Th} a local fine grid error indicator ηκ and a local two-grid error indicator ξκ. As noted above, cf. also Chapter 3, the local fine grid error indicator ηκ is analogous to the one that arises within the analysis of the standard IP DGFEM discretisation. With this in mind, ηκ represents the error arising from the linear fine grid solve defined in (3.11), while the local two-grid error indicator ξκ represents the error stemming from the approximation of the nonlinear coefficient µ(|∇uh,k|) on the fine mesh Th by the same quantity evaluated with respect to the coarse grid solution uH,K, i.e., the error committed by replacing µ(|∇uh,k|) by µ(|∇uH,K|). With this observation, we design the fine finite element space V (Th, k) by employing the local fine grid error indicators (3.20), while the coarse finite element space V (TH, K) is constructed in such a manner as to control the size of the local two-grid error indicators (3.21).

Assuming we have a method for selecting the elements to refine in the fine and coarse grid based on ηκ and ξκ, respectively, we can then devise the following general hp-refinement algorithm for the proposed two-grid method.

Algorithm 4.1. The finite element spaces V (Th, k) and V (TH, K) are constructed, based on

employing the following algorithm.

1. Initial step: Select the initial coarse and fine meshesTHandTh, as well as the initial coarse and fine polynomial degree distributions K and k, respectively, in such a manner that the resulting coarse and fine hp-finite element spaces V (TH, K) and V (Th, k), respectively,

satisfy the condition: V (TH, K)⊆ V (Th, k).

2. Select elements inTh andTH for refinement/derefinement, based on the local fine grid error indicators ηκ and the local two-grid error indicators ξκ from (3.20) and (3.21),

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respectively.

3. For elements marked for refinement in the fine and coarse mesh, determine whether to perform h- or p-refinement; see, for example, Fankhauser et al. [83], Houston & S ¨uli [116], Mitchell & McClain [142,143], Wihler [172,173].

4. Perform mesh smoothing to ensure:

• For all κ ∈ Ththere exists a coarse mesh element κH ∈ TH such that κ⊆ κH; • For all κ ∈ Thand κH ∈ TH, where κ⊆ κH, that KκH ≤ kκ.

In this thesis we perform h-refinement on the fine mesh Th and p-derefinement on the coarse meshTH where necessary.

Remark 4.1. For the purposes of the numerical experiments in the following section we

start the two-grid hp-adaptive algorithm with V (TH, K) = V (_{Th, k) in}Step 1above. In order to employ this algorithm we need a strategy to select elements inThandTH for refinement/derefinement, cf.Step 2above. In this chapter we propose two different methods.

4.2.1 Strategy One - Independent Fine and Coarse Grid Refinement

We base the first refinement strategy on the principal that we want the local two-grid error indicators to always be less than the local fine grid error indicators. We refine the fine grid as for a standard method, using the local fine grid error indicators ηκ, and then refine the coarse grid wherever ξκ&ηκ; cf.Figure 4.2for a graphical demonstration of the refinement routine.

Algorithm 4.2. Elements in the coarse and fine meshesTh andTH, respectively, are selected for refinement/derefinement based on employing the following algorithm.

1. Select fine grid elements: Using the fine grid error indicators ηκ from (3.20), apply a

standard refinement strategy to mark fine grid elements with a comparatively large error contribution for refinement, e.g. using the fixed fraction refinement strategy.

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Elements with largest ηκ

(a) Fine Grid

Elements with λξκ≥ηκ

(b) Coarse Grid

Figure 4.2: One step of theAlgorithm 4.2refinement strategy, demonstrating how the fine(a)

and coarse(b)meshes are refined independently for h-refinement; we note p-refinement works

in an analogous manner.

2. Select coarse grid elements: for a fixed constant steering parameter 0≤ λ < ∞, for each

element κ∈ Th, if λξκ ≥ ηκthen mark for refinement the coarse element κH ∈ TH where κ_{⊆ κH}.

Remark 4.2. We note that the algorithm allows the steering parameter λ to be zero. In

this situation no coarse mesh refinement will be performed and hence the algorithm will only refine the fine mesh.

4.2.2 Strategy Two - Linked Fine and Coarse Grid Refinement

An issue with the refinement strategy outlined in the previous section is that it always performs coarse refinement if the local two-grid error indicator is larger than the local fine grid error indicator, regardless of how much this error contributes to the total er- ror. As an example, if the local two-grid error indicator ξκ on one element κ is greater than the local fine grid error indicator ηκ on κ, but ξκ is several orders of magnitude lower than the error indicators on the other elements present in the mesh Th it will

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ξκ≥λCηκ ηκ≥λFξκ Both

Fine Grid:

Coarse Grid:

Figure 4.3: One step of theAlgorithm 4.3refinement strategy, demonstrating how the elements

selected for possible refinement based on ηκ+ ξκin the fine grid are used to decide on how to

perform fine and coarse mesh h-refinement; we note p-refinement works analogously.

still be refined, which will probably result in a minimal reduction of the overall error. The second strategy we propose, therefore, only considers elements based on the total local error indicator (ηκ+ ξκ) and then decides whether to perform coarse or fine refine- ment based on if the local two-grid error indicator or the local fine grid error indicator, respectively, is largest; cf. Figure 4.3for a graphical demonstration of the refinement routine.

Algorithm 4.3. Elements in the coarse and fine meshesTh andTH, respectively, are selected for refinement/derefinement based on employing the following algorithm.

1. Determine the sets R(Th) _{⊆ Th} and D(Th) _{⊆ Th} of fine elements to be (potentially) refined/derefined, respectively, based on the size of ηκ+ ξκ using a standard refinement

algorithm, e.g., the fixed fraction refinement strategy.

2. For all elements selected for derefinement decide whether to perform derefinement of the fine or coarse mesh: for all κ∈ D(Th)

• if λFξκ ≤ ηκ derefine the coarse element κH ∈ TH, where κ⊆ κH, and • if λCηκ ≤ ξκderefine the fine element κ.

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3. For all elements selected for refinement decide whether to perform refinement of the fine or coarse mesh: for all κ∈ R(Th)

• if λFξκ ≤ ηκ refine the fine element κ and

• if λCηκ ≤ ξκrefine the coarse element κH ∈ TH, where κ⊆ κH.

Here, λF, λC ∈ (0, ∞) are steering parameters selected such that λFλC ≤ 1.

Remark 4.3. We note that it is possible for a coarse element κH ∈ TH to be marked for both refinement and derefinement. When this occurs the coarse element is refined, as refinement should take precedence over derefinement.

Proposition 4.1. For all elements κ ∈ R(Th) either the fine element κ ∈ Th or the coarse element κH ∈ TH, where κ⊆ κH, will be marked for refinement.

Proof. To prove this statement it is sufficient to show that either p(κ) : λFξκ ≤ ηκ or q(κ) : λCηκ ≤ ξκ is true for all κ ∈ Th. For any κ ∈ Th, if p(κ) is true then p(κ)∨ q(κ) is true by definition; hence, it is only necessary to prove that q(κ) is true if p(κ) is false. As q(κ) is false and λFλC ≤ 1 then

λFξκ> ηκ≥ λFλCηκ.

Dividing through by λF > 0 gives that ξκ ≥ λCηκ; hence, q(κ) is true if p(κ) is false.

Remark 4.4. We note that although a similar result exists for the derefinement of ele-

ments κ∈ D(Th) it is possible for no element to be derefined for an element κ∈ D(Th) due toRemark 4.3.