# Resolving Boundary Layers with Harmonic Extension Finite Elements

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Boundary Layers

**Definition**

**1**

**Definition**

**2**

#### 2.2. Adaptive Reference Elements

**Definition**

**3.**(Planar adaptive reference element (ARE))

#### 2.2.1. Shape Functions

#### 2.2.2. Type of Reference Element

#### 2.2.3. Mesh Generation and Refinement

#### 2.3. Model Problems

#### 2.3.1. Reaction-Diffusion Problem

#### 2.3.2. Cylindrical Shells

## 3. Boundary Layer Resolution

## 4. Computational Asymptotic Analysis

#### 4.1. Solving Parameter-Dependent Sequences of Linear Systems

**Remark**

**1.**

#### 4.2. Recovering Quantities of Interest

## 5. Numerical Experiments

#### 5.1. Reaction–Diffusion

#### 5.2. Pitkäranta Cylinder

#### 5.2.1. On Numerical Locking

#### 5.2.2. Convergence Results

#### 5.2.3. Energy Dependence

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Solutions of model problems. In elasticity problems, the strongest boundary layers may occur in other vector field components. (

**a**) Reaction–diffusion. (

**b**) Pitkäranta Cylinder. Detail of transverse deflection (one quarter of the cylinder unfolded).

**Figure 2.**Adaptive reference element A: Quadrilateral with five nodes. Minimal implementation mesh with three triangles. The element mapping is defined using the four corner nodes only.

**Figure 5.**Boundary layer mesh example. Four levels of elements toward the right hand boundary. Normalised AEs are given as node lists, every split edge has the same parametrisation on the reference element. Types are labelled in the order of occurrence. (

**a**) Boundary layer mesh. (

**b**) Uniform: List of AEs and AREs (types). The node identifiers 6 and 13 have been swapped and hence all three labelled elements have different types. (

**c**) Non-uniform: $\left(6\right)\leftrightarrow \left(13\right)$: List of AE and ARE (types).

**Figure 6.**Sequence of refined meshes. Same replacement rule applied three times with a unit square as the initial configuration. The underlying finite element discretisation is continuous, despite the apparent hanging nodes. (

**a**) Rule applied once. (

**b**) Rule applied twice. (

**c**) After three applications of the rule.

**Figure 7.**Cylindrical shell: Layer structure. Examples of minimal meshes for a case with a generator at $x=1$ with $t=1/100$. (

**a**) Layer structure for the axial directions, there also exists a possible smooth component spanning the entire domain. (

**b**) Mesh adapted for $\sqrt{t}$. (

**c**) Mesh adapted for $\sqrt{t}$ and t.

**Figure 8.**Reaction–diffusion with $\u03f5=1/100$. Convergence graph has a characteristic staircasing behaviour, where the even nature of the exact solution over the domain results in no convergence as the polynomial order changes from even to odd, $p=2,\dots ,8$. (

**a**) Symmetric mesh. (

**b**) ${L}^{2}$-convergence (absolute error).

**Figure 9.**Pitkäranta cylinder: clamped case. Detail of the domain with symmetry/antisymmetry boundary conditions applied. The boundary layer in the rotation component with the characteristic length scale of $\sqrt{t}=1/10$ is clearly visible. (

**a**) Transverse deflection. (

**b**) Rotation $\theta $.

**Figure 10.**Pitkäranta cylinder, $t=1/100$. (

**a**) Clamped case on $10\times 10$-grid: Mesh. (

**b**) Free boundary: Mesh. (

**c**) Clamped case on $10\times 10$-grid: Errors. Squared energy norm convergence (absolute error) in $p=1,\dots ,4$ using different discretisations. (

**d**) Free case: Errors. Squared energy norm convergence (absolute error) in $p=1,\dots ,4$ using different discretisations. (

**e**) Clamped case on $g\times g$-grid: Errors. Squared energy norm convergence (absolute error) at $p=2$ over a series of uniform discretisations. (

**f**) Free case on $g\times g$-grid: Errors. Squared energy norm convergence (absolute error) at $p=2$ over a series of uniform discretisations. N is the number of degrees of freedom.

**Figure 11.**ARE convergence of Pitkäranta cylinder, $t=1/100$. Squared energy norm convergence (absolute error) in $p=2,\dots ,8$ using different discretisations: ARE 1: one layer, ARE 2: two layers, P: full tensor product grid with two layers, PF: full tensor product grid with one layer, PO: minimal full tensor product grid conforming to ARE 1. The observed convergence of the proposed method in both test cases agrees with the standard p-version. N is the number of degrees of freedom. (

**a**) Clamped case. (

**b**) Free boundary.

**Figure 12.**Clamped case. Practically perfect agreement with the a priori predictions. Notice, that ${\kappa}_{22}$ is essentially constant. Additionally, ${\rho}_{1}\to 0$ indicates that the Kirchhoff–Love condition is satisfied without imposing it within the model explicitly. (

**a**) ${\kappa}_{11}\sim 1/t$. (

**b**) ${\kappa}_{22}\sim 1$. (

**c**) ${\rho}_{1}\sim t$.

**Figure 13.**Free boundary. Notice, that ${\rho}_{1}$ is diverging, however, with a very small constant. This indicates that the boundary layer resolution is not accurate enough to conform to Kirchhoff–Love condition. The aggregate parameter-dependence is correct, the bending terms dominate the energy completely. (

**a**) ${\kappa}_{11}\left(\mathbf{u}\right)\sim 1/{t}^{2}$. (

**b**) ${\kappa}_{22}\left(\mathbf{u}\right)\sim 1/{t}^{2}$. (

**c**) ${\rho}_{1}\left(\mathbf{u}\right)\sim 1/\sqrt{t}$.

**Table 1.**Reference values used in numerical experiments. Reaction–diffusion with $\u03f5=1/100$ and Pitkäranta cylinder with $t=1/100$.

Case | Norm | Value Used | Exact |
---|---|---|---|

Reaction-Diffusion | ${L}^{2}$ | 0.83673056017854 | $\frac{\sqrt{\frac{1}{10}\left(13+40{e}^{10}+7{e}^{20}\right)}}{1+{e}^{10}}$ |

Cylinder (Clamped) | Squared energy | 2.6882879572571783 | - |

Cylinder (Free) | Squared energy | 7043.3120530934690 | - |

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Hakula, H.
Resolving Boundary Layers with Harmonic Extension Finite Elements. *Math. Comput. Appl.* **2022**, *27*, 57.
https://doi.org/10.3390/mca27040057

**AMA Style**

Hakula H.
Resolving Boundary Layers with Harmonic Extension Finite Elements. *Mathematical and Computational Applications*. 2022; 27(4):57.
https://doi.org/10.3390/mca27040057

**Chicago/Turabian Style**

Hakula, Harri.
2022. "Resolving Boundary Layers with Harmonic Extension Finite Elements" *Mathematical and Computational Applications* 27, no. 4: 57.
https://doi.org/10.3390/mca27040057