ON EXPLICIT BOUNDS IN THE ERROR FOR THE $ mathrm{H_o^1} $-PROJECTION INTO PIECEWISE POLYNOMIAL SPACES

The values of constants appearing in error estimates of approximations by finite element methods play an important role in numerical verification methods for elliptic equations (Nakao and Yamamoto(1998),Yamamoto and Nakao(1995) , etc.).For efficient implementation of the verification algorithms on computers, it is necessary that these constants can be estimated as close as possible to their optimal values. In Nakao,Yamamoto and Kimura(1998), the optimal constant was derived for quadratic elements as well as a nearly optimal value for cubic elements. In this paper, we establish a method to calculate the values of constants for approximation by piecewise polynomials of arbitrary degree and to give bounds on the difference between the constants so calculated and optimal values.