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

<table>Condition number of <svg height="13.675" id="M258" 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">
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</svg> depending on <svg height="14.5625" id="M259" 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">
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<tspan style="font-size: 8.75px; " x="0" y="0">7</tspan>
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</svg> and <svg height="14.7125" id="M260" 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">
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<g transform="translate(72,-60.23)">
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<tspan style="font-size: 8.75px; " x="0" y="0">1</tspan>
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</svg>. (a) Grid spacing <svg height="11.0125" id="M261" 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">
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<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>
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</svg>; (b) Grid spacing <svg height="11.0125" id="M262" 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">
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<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>
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Advances in Numerical Analysis

fig9

Figure 9

Figure 9: A Family of Sixth-Order Compact Finite-Difference Schemes for the Three-Dimensional Poisson Equation 