A Family of Sixth-Order Compact Finite-Difference Schemes for the Three-Dimensional Poisson Equation

<table>Computational accuracies for the first <svg height="11.1" id="M272" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 15.75 11.1" width="15.75" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.1)">
<g transform="translate(72,-63.12)">
<text transform="matrix(1,0,0,-1,-71.95,63.35)">
<tspan style="font-size: 12.50px; " x="0" y="0">2</tspan>
<tspan style="font-size: 12.50px; " x="6.2515001" y="0">5</tspan>
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</svg> eigenvalues of (<a href="https://static.hindawi.com/articles/ana/volume-2010/352174/figures/#EEq25">6.1</a>) for different choices of <svg height="14.5625" id="M273" style="vertical-align:-3.20526pt" version="1.1" viewbox="0 0 14.9375 14.5625" width="14.9375" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.5625)">
<g transform="translate(72,-60.35)">
<text transform="matrix(1,0,0,-1,-71.95,63.6)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝛽</tspan>
</text>
<text transform="matrix(1,0,0,-1,-64.97,60.47)">
<tspan style="font-size: 8.75px; " x="0" y="0">7</tspan>
</text>
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</svg>  and   <svg height="14.7125" id="M274" style="vertical-align:-3.32816pt" version="1.1" viewbox="0 0 20.4125 14.7125" width="20.4125" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,14.7125)">
<g transform="translate(72,-60.23)">
<text transform="matrix(1,0,0,-1,-71.95,63.6)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝛽</tspan>
</text>
<text transform="matrix(1,0,0,-1,-64.97,60.47)">
<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
<tspan style="font-size: 8.75px; " x="4.3759999" y="0">9</tspan>
</text>
</g>
</g>
</svg> for <svg height="13.675" id="M275" style="vertical-align:-2.20705pt" version="1.1" viewbox="0 0 12.2875 13.675" width="12.2875" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,13.675)">
<g transform="translate(72,-61.06)">
<text transform="matrix(1,0,0,-1,-71.95,63.31)">
<tspan style="font-size: 12.50px; " x="0" y="0">𝐐</tspan>
</text>
</g>
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</svg>. (a) Grid spacing <svg height="11.0125" id="M276" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 56.325001 11.0125" width="56.325001" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.0125)">
<g transform="translate(72,-63.19)">
<text transform="matrix(1,0,0,-1,-71.95,63.41)">
<tspan style="font-size: 12.50px; " x="0" y="0">ℎ</tspan>
<tspan style="font-size: 12.50px; " x="10.677562" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="22.717951" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="28.969452" y="0">/</tspan>
<tspan style="font-size: 12.50px; " x="32.445286" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="38.696785" y="0">0</tspan>
</text>
</g>
</g>
</svg>; (b) Grid spacing <svg height="11.0125" id="M277" style="vertical-align:-0.17555pt" version="1.1" viewbox="0 0 60.224998 11.0125" width="60.224998" xmlns="http://www.w3.org/2000/svg">
<g transform="matrix(1.25,0,0,-1.25,0,11.0125)">
<g transform="translate(72,-63.19)">
<text transform="matrix(1,0,0,-1,-71.95,63.41)">
<tspan style="font-size: 12.50px; " x="0" y="0">ℎ</tspan>
<tspan style="font-size: 12.50px; " x="10.677562" y="0">=</tspan>
<tspan style="font-size: 12.50px; " x="22.717951" y="0">1</tspan>
<tspan style="font-size: 12.50px; " x="28.969452" y="0">/</tspan>
<tspan style="font-size: 12.50px; " x="32.445286" y="0">2</tspan>
<tspan style="font-size: 12.50px; " x="38.696785" y="0">0</tspan>
<tspan style="font-size: 12.50px; " x="44.948284" y="0">.</tspan>
</text>
</g>
</g>
</svg></table>

Advances in Numerical Analysis

fig10

Figure 10

Figure 10: A Family of Sixth-Order Compact Finite-Difference Schemes for the Three-Dimensional Poisson Equation 