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.
"Numerical Verification Method of Solutions for Elliptic Equations and Its Application to the Rayleigh--Bénard Problem." Japan J. Indust. Appl. Math. 26 (2-3) 443 - 463, October 2009.