We first summarize the general concept of our verification method of solutions for elliptic equations. Next, as an application of our method, a survey and future works on the numerical verification method of solutions for heat convection problems known as Rayleigh-Bénard problem are described. We will give a method to verify the existence of bifurcating solutions of the two-dimensional problem and the bifurcation point itself. Finally, an extension to the three-dimensional case and future works will be described.

中文翻译：

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椭圆方程解的数值验证方法及其在Rayleigh-Bénard问题中的应用

我们首先总结我们椭圆方程解的验证方法的一般概念。接下来，作为我们方法的应用，描述了对称为 Rayleigh-Bénard 问题的热对流问题解的数值验证方法的调查和未来工作。我们将给出一种方法来验证二维问题的分岔解和分岔点本身的存在性。最后，将描述对 3D 案例的扩展和未来的工作。