%0 Journal Article
%T 抛物方程的变网格连续时空有限体积元法<br>Variable Mesh Continuous Space-Time Finite Volume Element Method for Parabolic Equations
%A 肖宇宇
%A 何斯日古楞
%A 陈娟
%J Advances in Applied Mathematics
%P 148-163
%@ 2324-8009
%D 2025
%I Hans Publishing
%R 10.12677/aam.2025.146308
%X 从三次Lagrange插值最佳应力点的理论出发&#65292;本文提出一种变网格连续时空有限体积元格式&#65292;旨在解决抛物方程数值求解问题。通过耦合Legendre-Lobatto节点的Lagrange插值多项式与Gauss积分准则&#65292;在时空非匹配网格剖分条件下严格证明了数值解的存在唯一性&#65292;并建立<math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msup>
   <mi>L</mi>
   <mi>&#x221E;</mi>
  </msup>
  <mrow><mo>(</mo>
   <mrow>
    <msup>
     <mi>L</mi>
     <mn>2</mn>
    </msup>
    </mrow>
  <mo>)</mo></mrow></mrow>
</math>

和<math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msup>
   <mi>L</mi>
   <mi>&#x221E;</mi>
  </msup>
  <mrow><mo>(</mo>
   <mrow>
    <msup>
     <mi>H</mi>
     <mn>1</mn>
    </msup>
    </mrow>
  <mo>)</mo></mrow></mrow>
</math>

的最优阶误差估计理论。数值实验结果表明收敛数据与理论预测高度一致&#65292;验证了算法在非均匀网格环境中的计算优势与理论分析的有效性。<br>Based on the theoretical framework of cubic Lagrange interpolation with optimal stress nodes, this study develops a variable mesh continuous space-time finite volume element scheme to resolve numerical challenges in solving parabolic equations. By integrating Lagrange interpolation polynomials at Legendre-Lobatto nodes with Gauss quadrature rules, we rigorously prove the existence and uniqueness of numerical solutions under spacetime non-matching grid partitions. Optimal-order error estimates in <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msup>
   <mi>L</mi>
   <mi>&#x221E;</mi>
  </msup>
  <mrow><mo>(</mo>
   <mrow>
    <msup>
     <mi>L</mi>
     <mn>2</mn>
    </msup>
    </mrow>
  <mo>)</mo></mrow></mrow>
</math>

 and <math xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <msup>
   <mi>L</mi>
   <mi>&#x221E;</mi>
  </msup>
  <mrow><mo>(</mo>
   <mrow>
    <msup>
     <mi>H</mi>
     <mn>1</mn>
    </msup>
    </mrow>
  <mo>)</mo></mrow></mrow>
</math>

 norms are theoretically established. Numerical experiments demonstrate excellent agreement between convergence rates and theoretical predictions, confirming the computational advantages of the proposed algorithm in non-uniform grid environments and the validity of theoretical analysis.
%K 抛物方程&#65292
%K 变网格连续时空元&#65292
%K 有限体积元法&#65292
%K 误差估计<br>Parabolic Equations
%K Variable Mesh Continuous Space-Time Elements
%K Finite Volume Element Method
%K Error Estimates
%U http://www.hanspub.org/journal/PaperInformation.aspx?PaperID=117584