# A multigrid method for nonconforming FE-discretisations with application to non-matching grids

## Summary (2 min read)

### 1. Introduction

- Recently, domain decomposition methods have been applied to situations where subdomain meshes may be separately constructed and are non-matching along the interfaces.
- When multigrid methods are designed, there is now the problem that the finite element spaces are not nested.
- Therefore, the authors have to construct appropriate prolongation operators.
- The approximation property for the convergence proof will be derived from an auxiliary problem.
- After completing the paper, the authors learnt about the paper [11] investigating the mortar finite element method by other theoretical means.

### 3. Construction of the Prolongation

- For the sake of simplicity, the authors omit these indices.
- V` plays an important role when Stevenson [14] considers an axiomatic framework for the Cascadic multigrid algorithms suitable for nonconforming elements.

### 3.2. Conditions on π and σ

- The space 6 will be equipped with the norm and the scalar product of H0, and S is assumed to become a Hilbert space by a norm ‖·‖S and a scalar product (·, ·)S whose specification may depend on the specific finite element space.
- The introduction of the intermediate space S gives us more freedom in the construction of the mappings.
- The authors emphasise that only boundedness in H0 is required for ι, while the concept of Brenner [8, 15] also refers to conditions with respect to the energy norm.

### 4. Coarse-Grid Correction and Approximation Property

- The coarse-grid correction e`−1 ∈ V`−1 approximates the finite element function e` ∈ V`.
- Note that it is required for converting the function w`−1 from V`−1 into an element of V`.

### 4.2. An Auxiliary Problem

- To this end the authors introduce two Riesz representations of the residue.
- After dividing by ‖g‖0 and inserting the estimate of r̄` above, the authors obtain the required inequality.

### 4.3. Approximation Property

- The error enew` = φ` enew` from (4.5) after the coarse-grid correction will be estimated in the following proposition.
- After inserting this estimate into (4.11) the proof is complete.

### 5. Smoothing Property and Multigrid Convergence

- 1. Smoothing Property Specifically, the authors assume that the matrix.
- The cases of A` not being positive definite or of other smoothing iterations are described in [12], too.

### 5.2. Multigrid Convergence

- The iteration matrix of the two-grid iteration (with pre-smoothing, only) equals MTGM` (ν) = (A−1` − pA−1`−1 r)(A` Sν` ), where ν is the number of smoothing iterations described by the iteration matrix S`.
- Replacing the exact solution of the coarse-grid problem by two iterations of the multigrid method, the authors obtain the W-cycle.
- Since Lemma 5.1 implies (5.6), the multigrid convergence is proved.

### 6. Application to Non-Matching Grids

- The authors describe and analyse a multigrid method for solving systems of algebraic equations arising from a finite element method based on non-matching triangulations.
- The discretisation is done by the mortar technique, see [2, 3].
- A multigrid method is presented and analysed which makes use of the general scheme presented in the previous sections.

### 6.1. Discrete Problem

- In the mortar method nomenclature, this case is called geometrically conforming.
- In particular, at a cross point there are several nodal points (xp, j(p)) with identical position xp but different j(p).
- To define suitable spaces for discretisation of (6.1) the authors need to impose some constraints on the jumps of functions from X`( ) on 0ij which are called mortar 1 I maps the vertex into the index of the related subdomain.
- Similarly, one considers the mesh T`,j ∩ δ̄m on δm.

### 6.3. Choice of 6, S, σ, and π

- In the diagram below, V`−1 and V` are the mortar spaces introduced in Section 6.1.1.
- After a slight modification at nodal points neighboured to the slave nodes, one obtains zero coefficients for all p ∈ Because of the Lagrange basis property, φ`v` recovers the function v` ∈ V`.
- Hence, if the underlying second order boundary value problem satisfies the solvability and regularity conditions, all necessary requirements for the two- and multigrid convergence are satisfied.

### 7. Geometrically Nonconforming Case

- This case is called the geometrically nonconforming case in the mortar method, and a typical situation is depicted in Fig 1.
- The authors first formulate a discrete problem for (6.1) and then extend their treatment of the geometrically conforming case to this one.

### 7.1. Discrete Problem

- To formulate the discrete problem, the authors will adapt some of their previous notations and introduce some new ones.
- The edges 0j,k which are not masters are called slaves and denoted by δm, m ∈M.
- The estimates of the consistency errors are local in nature and carry over without changes.
- The authors goal is to design and analyse the multigrid method for solving (7.3) using the previous scheme.

### 7.2. Matrix Form

- The basis functions b`,p are defined again by the Lagrange basis property similar to those in Section 6.
- A` is symmetric and positive definite and its eigenvalues satisfy also the inequalities (6.10).

### 7.3. Two-Grid and Multigrid Method

- This is sufficient to analyse the multigrid method.
- Since ` is a variable level number, the space V`−1 and the mortar projection5`−1,m are defined as well.
- Using this lemma, the authors check that Lemmata 6.2, 6.4 and 6.5 from Section 6 are also valid for the discussed case.

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##### Citations

^{1}, University of Bath

^{2}, Max Planck Society

^{3}, University of Zurich

^{4}

86 citations

77 citations

### Additional excerpts

...We define φ, following ideas from [32]....

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### Cites background or methods from "A multigrid method for nonconformin..."

...Proof: One can use the same technique which has been presented for the case of a scalar elliptic problem in [4]....

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...In [4], the mapping iu has been assumed to be the product iu 1⁄4 p r with r : Rl ! S and p : S ! Vl to allow more flexibility in constructing a suitable iu....

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...The general framework of analysing the two-level convergence of non-nested multi-grid methods, developed in [4] for elliptic problems, will be the starting point for the methods studied herein....

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### Cites methods from "A multigrid method for nonconformin..."

...Applying the elegant general theory developed in [ 5 ], which builds upon the classical results in [2], we give in Sect . 6 a convergence proof for the W-cycle....

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...We establish the convergence of the multigrid iteration with the aid of the theory in [ 5 ], which became available just in time for our application....

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##### References

2,545 citations

### "A multigrid method for nonconformin..." refers background in this paper

...The cases of A` not being positive definite or of other smoothing iterations are described in [12], too....

[...]

998 citations

851 citations

### "A multigrid method for nonconformin..." refers background or methods in this paper

...The method was called mortar element method in [3]....

[...]

...A proof of the error estimate in H 1() can be found in [2] and [3] and the L2 estimate is also given in [1] without a proof....

[...]

...The discretisation is done by the mortar technique, see [2, 3]....

[...]

705 citations

193 citations

### "A multigrid method for nonconformin..." refers background or methods in this paper

...A proof of the error estimate in H 1() can be found in [2] and [3] and the L2 estimate is also given in [1] without a proof....

[...]

...The discretisation is done by the mortar technique, see [2, 3]....

[...]