On error estimation of finite element approximations to the elliptic equations in nonconvex polygonal domains

Numerical verification methods, so-called Nakao's methods, on existence or uniqueness of solutions to PDEs have been developed by Nakao and his group including the authors. They are based on the error estimation of approximate solutions which are mainly computed by FEM.It is a standard way of the error estimation of FEM to estimate the projection errors by elementwise interpolation errors. There are some constants in the error estimation, which depend on the mesh size parameters h. The explicit values of the constants are necessary in order to use Nakao's method. However, there were not so many researches for the computation of the explicit values of the constants. Then we had to develop the computation by ourselves, especially with guaranteed accuracy. Note that the methods of the computation depend on the dimension, the degree of bases, and the shape of the domain, etc.The present paper shows how we have developed the methods to calculate the constants and describes new results for nonconvex domains.