# Modified Representations for the Close Evaluation Problem

## Abstract

**:**

## 1. Introduction

## 2. Motivation for Modified Representations

## 3. Modified Representations

#### 3.1. Modified Representation for the Laplace Double-Layer Potential

**Proposition**

**1.**

#### 3.2. Modified Representation for the Laplace Single-Layer Potential

**Proposition**

**2.**

- The linear function $\mathtt{v}\left(y\right)={n}_{{x}^{*}}\xb7y$;
- The function $\mathtt{v}\left(y\right)={2}^{d-1}\pi G(y,{x}^{*}+{n}_{{x}^{*}})$ based on Green’s function;
- The quadratic product function $\mathtt{v}\left(y\right)=\frac{({y}_{1}-{x}_{0,1})({y}_{2}-{x}_{0,2})}{{n}_{{x}^{*},1}({x}_{2}^{*}-{x}_{0,2})+{n}_{{x}^{*},2}({x}_{1}^{*}-{x}_{0,1})}$, ${x}_{0}\in D$;
- The quadratic difference function $\mathtt{v}\left(y\right)=\frac{1}{2}\frac{{({y}_{1}-{x}_{0,1})}^{2}-{({y}_{2}-{x}_{0,2})}^{2}}{{n}_{{x}^{*},1}({x}_{1}^{*}-{x}_{0,1})-{n}_{{x}^{*},2}({x}_{2}^{*}-{x}_{0,2})}$, ${x}_{0}\in D.$

#### 3.3. Modified Representation for the Helmholtz Double- and Single-Layer Potentials

**Proposition**

**3.**

## 4. Numerical Examples

#### 4.1. Exterior Neumann Laplace Problem

#### 4.1.1. Example 1: Exterior Laplace in Two Dimensions

**V0:**standard representation (7);**V1:**modified representation (17) with the linear function ${\mathtt{v}}_{1}\left(y\right)={n}_{{x}^{*}}\xb7y;$**V2:**modified representation (17) with the Green’s function ${\mathtt{v}}_{2}\left(y\right)=2\pi G(y,{x}^{*}+{n}^{*});$**V3:**modified representation (17) with the quadratic function ${\mathtt{v}}_{3}\left(y\right)=\frac{1}{2}\frac{{y}_{1}^{2}-{y}_{2}^{2}}{{n}_{{x}^{*},1}{x}_{1}^{*}-{n}_{{x}^{*},2}{x}_{2}^{*}};$**V4:**modified representation (17) with the quadratic function$${\mathtt{v}}_{4}\left(y\right)=\frac{({y}_{1}-5)({y}_{2}-5)}{{n}_{{x}^{*},1}({x}_{2}^{*}-5)+{n}_{{x}^{*},2}({x}_{1}^{*}-5)}.$$

#### 4.1.2. Example 2: Exterior Laplace in Three Dimensions

**V0:**standard representation (7);**V1:**modified representation (17) with the linear function ${\mathtt{v}}_{1}\left(y\right)={n}_{{x}^{*}}\xb7y$;**V2:**modified representation (17) with the Green’s function ${\mathtt{v}}_{2}\left(y\right)=4\pi G(y,{x}^{*}+{n}^{*});$**V3:**modified representation (17) with the quadratic function ${\mathtt{v}}_{3}\left(y\right)=\frac{1}{2}\frac{{y}_{1}^{2}-{y}_{2}^{2}}{{n}_{{x}^{*},1}{x}_{1}^{*}-{n}_{{x}^{*},2}{x}_{2}^{*}};$**V4:**modified representation (17) with the quadratic product function$${\mathtt{v}}_{4}\left(y\right)=\frac{({y}_{1}-5)({y}_{2}-5)}{{n}_{{x}^{*},1}({x}_{2}^{*}-5)+{n}_{{x}^{*},2}({x}_{1}^{*}-5)}.$$

#### 4.2. Scattering Problem

#### 4.2.1. Example 3: Scattering in Two Dimensions

**V0:**standard representation (19);

#### 4.2.2. Example 4: Scattering in Three Dimensions

#### 4.2.3. High Frequency Behavior

## 5. Modified Boundary Integral Equations

**Proposition**

**4.**

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Kress Product Quadrature

## Appendix B. Galerkin Approximation

## Appendix C. Proof of Modified Representations

#### Appendix C.1. Modified Double-Layer Potential (14)

#### Appendix C.2. Proof of Proposition 2

#### Appendix C.3. Proof of Propositions 3, 4

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**Figure 1.**Laplace 2D single-layer. Plots of ${log}_{10}$ of the error for the evaluation of the solution of (2) out of the kite domain defined by the boundary $y\left(t\right)=(cost+0.65cos(2t)-0.65,1.5sint)$, $t\in [0,2\pi ]$, for the Neumann data, $g={\partial}_{n}{u}_{\mathrm{exact}}$ with ${x}_{0}=(0.1,0.4)$, for representations V0, V1, V2, V3, V4 computed using PTR with $N=128$. Computations are made on a boddy-fitted grid with $N\times 200$ grid points.

**Figure 3.**Laplace 2D single-layer. Log-log plots of the errors with respect to N made in computing the solution at some distance ℓ along the normal from point A plotted as black ×’s in Figure 1.

**Figure 4.**Laplace 3D single-layer. Log-log plots of the errors with respect to ℓ made in computing the solution of (2) for the Neumann data, $g\left({x}^{*}\right)=-\frac{{n}_{{x}^{*}}\xb7({x}^{*}-{x}_{0})}{|{x}^{*}-{x}_{0}{|}^{3}}$ with ${x}_{0}=(0,0,0)$, outside of a sphere a radius 2, along the normal of point A = $(-0.0065,-0.0327,1.9997)$ (

**left**), of point B = $(-0.3526,-1.7728,0.8561)$ (

**right**).

**Figure 5.**Laplace 3D single-layer. Log-log plots of the errors with respect to N made in computing the solution (as described in Figure 4) at some distance ℓ along the normal from point B = $(-0.3526,-1.7728,0.8561)$.

**Figure 6.**Helmholtz 2D. Plots of ${log}_{10}$ of the error for the evaluation of the solution of (18) out of the star domain defined by the boundary $y\left(t\right)=(1+0.3cos5t)*(cost,sint)$, $t\in [0,2\pi ]$, for the Dirichlet data, $f\left({x}^{*}\right)=\frac{i}{4}{H}_{0}^{\left(1\right)}\left(15\right|{x}^{*}-{x}_{0}\left|\right)$ with ${x}_{0}=(0.2,0.8)$, for representations V0, V1, computed using PTR with $N=256$.

**Figure 7.**Helmholtz 2D. Log-log plots of the errors made in computing the solution along the normal of the three points A, B, C, plotted as black ×’s in Figure 6.

**Figure 8.**Helmholtz 2D. Log-log plots of the errors with respect to N made in computing the solution at some distance ℓ along the normal from point A plotted as black ×’s in Figure 6.

**Figure 9.**Helmholtz 3D. Log-Log of the error along the normal for the evaluation of the solution of (18) out of the ellipsoid parameterized by $y(s,t)=(2cos(t)sin(s),sin(t)sin(s),2cos(s\left)\right)$, $(s,t)\in [0,\pi ]\times [-\pi ,\pi ]$, for the Dirichlet data $f\left({x}^{*}\right)=\frac{1}{4}\frac{{e}^{i5|z-{x}_{0}|}}{|x-{x}_{0}|}$ with ${x}_{0}=(0.1,0.2,0.3)$: at point A = $(-0.7664,0.0607,1.8433)$ (top row), at point B = $(-0.0098,-0.0096,1.9999)$ (bottom row), for various N.

**Figure 10.**Helmholtz 3D. Log-plot of the maximum error for computing the solution as described in Figure 9 with $\partial D$ being the ellipsoid parameterized by $y(s,t)=(2cos(t)sin(s),sin(t)sin(s),2cos(s\left)\right)$, $(s,t)\in [0,\pi ]\times [-\pi ,\pi ]$, at some distance ℓ along the normal from point A= $(-0.7664,0.0607,1.8433)$.

**Figure 11.**Helmholtz 2D. Log-Log of the maximum error in computing the solution of Problem (18) as described in Section 4.2.1, with respect to the wavenumber k, for various number of quadrature points N.

**Figure 12.**Helmholtz 3D. Log-Log of the maximum error in computing the solution of Problem (18) as described in Section 4.2.2, with respect to the wavenumber k, for various number of quadrature points N.

**Figure 13.**Helmholtz 2D. Log-Log plot of the error along the normal for the solution of (18) out of the star domain defined by the boundary $y\left(t\right)=(1.55+0.4cos5t)*(cost,sint)$, $t\in [0,2\pi ]$, for the Dirichlet data, $f\left({x}^{*}\right)=\frac{i}{4}{H}_{0}^{\left(1\right)}\left(15\right|{x}^{*}-{x}_{0}\left|\right)$ with ${x}_{0}=(0.2,0.8)$, at the three points A, B, C plotted as black ×’s in Figure 6.

**Figure 14.**Helmholtz 3D. Log-Log plot of the error for the problem described in Figure 9 using $N=32$, and for the four representations (standard or modified, off and on boundary).

**Table 1.**Laplace 2D single-layer. CPU times (in seconds) for various number of quadrature points and representations. Times account for computing the solution at $N\times 12$ grid points ($\ell ={10}^{-k}$, $k=\u301a0,11\u301b$) on a body-fitted grid.

Method | V0 | V1 | V2 | V3 | V4 |
---|---|---|---|---|---|

$N=128$ | 0.014 | 0.044 | 0.055 | 0.045 | 0.05 |

$N=256$ | 0.056 | 0.07 | 0.112 | 0.08 | 0.081 |

$N=512$ | 0.12 | 0.192 | 0.263 | 0.2 | 0.19 |

**Table 2.**Laplace 3D single-layer. CPU times (in seconds) for various number of quadrature points and representations for computing the solution (as described in Figure 4) from points A and B, for $\ell ={10}^{-k}$, $k=\u301a0,11\u301b$.

Method | V0 | V1 | V2 | V3 | V4 |
---|---|---|---|---|---|

N = 8 | 0.028 | 0.029 | 0.032 | 0.031 | 0.046 |

N = 16 | 0.143 | 0.146 | 0.148 | 0.150 | 0.142 |

N = 24 | 0.352 | 0.344 | 0.346 | 0.35 | 0.356 |

**Table 3.**Helmholtz 2D. CPU times (in seconds) for various number of quadrature points and representations. Times account for computing the solution for $N\times 12$ grid points (for $\ell ={10}^{-k}$, $k=\u301a0,11\u301b$) on a body-fitted grid.

Method | $\mathit{N}=128$ | $\mathit{N}=256$ | $\mathit{N}=512$ |
---|---|---|---|

V0 | 0.18 | 0.27 | 0.71 |

V1 | 0.21 | 0.33 | 0.89 |

**Table 4.**Helmholtz 3D. CPU times (in seconds) for various number of quadrature points and representations. Times account for computing the solution from points A and B, for $\ell ={10}^{-k}$, $k=\u301a0,11\u301b$.

Method | $\mathit{N}=8$ | $\mathit{N}=16$ | $\mathit{N}=20$ |
---|---|---|---|

V0 | 0.027 | 0.15 | 0.313 |

V1 | 0.03 | 0.15 | 0.314 |

**Table 5.**Helmholtz 2D. CPU times (in seconds) for various number of quadrature points to compute the solution of the boundary integral equation.

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**MDPI and ACS Style**

Carvalho, C. Modified Representations for the Close Evaluation Problem. *Math. Comput. Appl.* **2021**, *26*, 69.
https://doi.org/10.3390/mca26040069

**AMA Style**

Carvalho C. Modified Representations for the Close Evaluation Problem. *Mathematical and Computational Applications*. 2021; 26(4):69.
https://doi.org/10.3390/mca26040069

**Chicago/Turabian Style**

Carvalho, Camille. 2021. "Modified Representations for the Close Evaluation Problem" *Mathematical and Computational Applications* 26, no. 4: 69.
https://doi.org/10.3390/mca26040069