# A Comparative Study of Several Classical, Discrete Differential and Isogeometric Methods for Solving Poisson’s Equation on the Disk

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## Abstract

**:**

## 1. Introduction

- a qualitative comparison between classical finite elements, a DDG approach and four isogeometric constructions;
- an investigation of quadrature formulas for subdivision IgA finite elements;
- implementation of an IgA method for ${C}^{1}$ functions on complex domains that is based on ${G}^{1}$ constructions and yields $O\left({h}^{3}\right)$ convergence, also at irregular points; this improved convergence is confirmed for an L-shaped domain and for an elastic plate with a circular hole;
- implementation of an IgA method with singular parameterization at irregular points that yields $O\left({h}^{3}\right)$ convergence also at irregular points.

#### Overview

## 2. Classical Finite and DDG Elements

#### 2.1. ${C}^{0}$ Quadratic Triangular Elements

**Figure 1.**${C}^{0}$ quadratic basis functions. (

**a**,

**c**) top view with height scale. (

**a**) Nodal basis function; (

**b**) BB-piece of (a); (

**c**) mid-edge basis function; (

**d**) BB-piece of (c).

#### 2.2. Hsieh–Clough–Tocher Elements

**Figure 2.**Hsieh–Clough–Tocher (HCT) basis functions (top view with height scale). (

**a**) ${b}_{3i}^{\u25b5}$: nodal basis function; (

**b**) ${b}_{3i+1}^{\u25b5}$: x-derivative basis function; (

**c**) ${b}_{3N+k}^{\u25b5}$: mid-edge normal derivative function.

#### 2.3. The Discrete Differential Geometry Approach

**Figure 3.**Discrete differential geometry (DDG) notation and linear functions. (

**a**) Notation of Equation (2); (

**b**) top view of the linear “hat” function.

## 3. The Isogeometric Approach

**Figure 4.**A basis function, ${b}_{i}$, is the composition of the basis function, ${b}_{i}^{\square}$, on the tensor-product parameter domain, T, and the inverse of the geometry mapping, $\mathbf{x}$. (

**a**) The union of $4\times 4$ domains T and its image, Ω, under $4\times 4$ maps ${\mathbf{x}}_{\alpha}$. (

**a**) (

**left**) Union domains T and (

**right**) the physical domain Ω; (

**b**) basis function ${b}_{i}^{\square}$ on T, respectively on Ω.

**Figure 5.**The non-smooth ${C}^{0}$ bi-3 basis function: (

**a**,

**left**) A quad mesh of points associated with B-spline-like functions, for $n=5$. (

**a**,

**right**) The coefficients of the patches in tensor-product BB-form defined by the ${C}^{2}$ extension of the regular spline complex towards the extraordinary point (see [29]). Note the $n=5$ points of valence three surrounding the central n-valent point. (

**a**) Control net and ${C}^{2}$ extension in BB-form; (

**b**) ${C}^{0}$ bi-3 basis function.

#### 3.1. Bi-3 Elements That Are ${C}^{0}$ at Extraordinary Points

#### 3.2. Catmull–Clark Elements

**Figure 6.**Catmull–Clark elements. (

**a**) Refinement Level 3; (

**b**) refinement Level 7; (

**c**) a Catmull–Clark subdivision function.

**Table 1.**Error (scaled by ${10}^{-5}$) of the computed solution of Poisson’s equation by Catmull–Clark subdivision on Disk 1 (see Figure Figure 6), for different levels of subdivision when applying the Gauss quadrature. The subdivision is localized to not refine the overall mesh.

Depth | ${L}^{2}$ | ${L}^{\infty}$ |
---|---|---|

3 | 893.063 | 476.26 |

5 | 100.44 | 81.193 |

7 | 70.395 | 47.004 |

9 | 70.073 | 43.992 |

#### 3.3. Higher-Order ${G}^{1}$ Elements

**Figure 7.**${G}^{1}$ element at an extraordinary point. (

**a**) Two ${G}^{1}$ bi-3/bi-5 basis functions; (

**b**) ${G}^{1}$ bi-3/bi-5 basis function with onepatch in BB-form lifted up.

#### 3.4. Polar ${C}^{1}$ Elements

**Figure 8.**Polar elements for polar configurations. (

**a**) Modeling with ${C}^{1}$ polar functions; (

**b**) a ${C}^{1}$ polar basis function.

## 4. Solving the Poisson Equation

## 5. Numerical Results and Comparison

#### 5.1. Correct Gauss Quadrature for Catmull–Clark Subdivision

#### 5.2. Convergence Rates

#### 5.3. Complexity

**Figure 9.**Three types of meshes specific to each of the three classes of methods. (

**d**) ${C}^{0}$ quadratic, HCT, DDG elements: 384, $384\times 4$, $384\times 16$ elements; (

**e**) bi-3 ${C}^{0}$, Catmull–Clark and ${G}^{1}$ bi-3/bi-5 elements: 120, $120\times 4$, $120\times 16$ elements; (

**f**) polar ${C}^{1}$ elements: 100, $100\times 4$, $100\times 16$ elements. Columns

**a**,

**b**,

**c**correspond to refinement by halving h, hence quadrupling the number of elements, i.e. to (

**a**) Disk 1; (

**b**) Disk 2; (

**c**) Disk 3;

**Figure 10.**Convergence comparison between methods. Note that the graphs are in log-scale (the triangle indicates the convergence exponent in log-scale) and that higher mesh density is to the left, as the mesh spacing on the abscissa decreases. (

**a**) Error in ${L}^{2}$; (

**b**) error in ${L}^{\infty}$.

**Figure 11.**Poisson’s equation on Disk 1: difference graphs between the exact solution and the computed solution. (

**a**) ${C}^{0}$, bi-3; (

**b**) ${G}^{1}$ bi-3/bi-5; (

**c**) Catmull–Clark (level = 7), bi-3; (

**d**) ${C}^{1}$ polar, bi-3.

#### 5.4. The ${G}^{1}$ bi-3/bi-5 Elements on the L-shape and on the Elastic-Plate-with-Hole

## 6. Conclusions

**Figure 12.**The ${G}^{1}$ bi-3/bi-5solution of Laplace’s equation on the L-shape. (

**a**) h-refinement of the L-shape; (

**b**) the difference between the exact solution and the computed solution; (

**c**) ${L}^{2}$-error; (

**d**) ${L}^{\infty}$-error.

**Figure 13.**The ${G}^{1}$ bi-3/bi-5elements on the elastic plate with a circular hole. (

**a**) h-refinement; (

**b**) contour plots of ${\sigma}_{xx}$; (

**c**) ${L}^{2}$-error; (

**d**) ${L}^{\infty}$-error.

## Acknowledgments

## Author Contributions

- a qualitative comparison between classical finite elements, a DDG approach and four isogeometric constructions;
- an investigation of quadrature formulas for subdivision IgA finite elements;
- implementation of an IgA method for ${C}^{1}$ functions on complex domains that is based on ${G}^{1}$ constructions and yields $O\left({h}^{3}\right)$ convergence, also at irregular points; this improved convergence is confirmed for an L-shaped domain and for an elastic plate with a circular hole;
- implementation of an IgA method with singular parameterization at irregular points that yields $O\left({h}^{3}\right)$ convergence also at irregular points.

## Conflicts of Interest

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Nguyen, T.; Karčiauskas, K.; Peters, J.
A Comparative Study of Several Classical, Discrete Differential and Isogeometric Methods for Solving Poisson’s Equation on the Disk. *Axioms* **2014**, *3*, 280-299.
https://doi.org/10.3390/axioms3020280

**AMA Style**

Nguyen T, Karčiauskas K, Peters J.
A Comparative Study of Several Classical, Discrete Differential and Isogeometric Methods for Solving Poisson’s Equation on the Disk. *Axioms*. 2014; 3(2):280-299.
https://doi.org/10.3390/axioms3020280

**Chicago/Turabian Style**

Nguyen, Thien, Keçstutis Karčiauskas, and Jörg Peters.
2014. "A Comparative Study of Several Classical, Discrete Differential and Isogeometric Methods for Solving Poisson’s Equation on the Disk" *Axioms* 3, no. 2: 280-299.
https://doi.org/10.3390/axioms3020280