We consider the first type boundary value problem of the heat equation in two space dimensions on special polygons with interior angles \(\alpha _{j}\pi \), \(j=1,2,\ldots,M\), where \(\alpha _{j}\in \{ \frac{1}{2},\frac{1}{3},\frac{2}{3} \} \). To approximate the solution we develop two difference problems on hexagonal grids using two layers with 14 points. It is proved that the given implicit schemes in both difference problems are unconditionally stable. It is also shown that the solutions of the constructed Difference Problem 1 and Difference Problem 2 converge to the exact solution on the grids of order \(O ( h^{2}+\tau ^{2} ) \) and \(O ( h^{4}+\tau ) \) respectively, where h and \(\frac{\sqrt{3}}{2}h \) are the step sizes in space variables \(x_{1}\) and \(x_{2}\) respectively and τ is the step size in time. Furthermore, theoretical results are justified by numerical examples on a rectangle, trapezoid and parallelogram.

我们认为在与内角特殊的多边形二维空间的热传导方程的第一类型的边界值问题\（\阿尔法_ {Ĵ} \ PI \） ，\（J = 1,2，\ ldots，男\） ，其中\（\ alpha _ {j} \ in \ {\ frac {1} {2}，\ frac {1} {3}，\ frac {2} {3} \} \）中。为了近似解决方案，我们在使用14点的两层六边形网格上开发了两个差分问题。证明了两个差分问题中给定的隐式方案都是无条件稳定的。还表明，构造的差分问题1和差分问题2的解收敛到阶为\（O（h ^ {2} + \ tau ^ {2}）\）和\（O （h ^ {4} + \ tau）\），其中h和\（\ frac {\ sqrt {3}} {2} h \）分别是空间变量\（x_ {1} \）和\（x_ {2} \）中的步长，而τ是其中的步长。时间。此外，通过在矩形，梯形和平行四边形上的数值示例可以证明理论结果是正确的。