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The present paper describes the result of the error estimation of a numerical solution to topology optimization problems of domains in which boundary value problems of partial differential equations are defined. In the previous paper, we formulated a problem using density as a design variable, presented a regular solution by using a gradient method in a function space, and called the method the H1 gradient method. The main result in this paper shows the first order convergence in the H1 norm of the solution in the H1 gradient method with respect to the size of the finite elements if first order elements are used for the design variable and the solution to the boundary value problem. The main result is confirmed by numerical examples. [ABSTRACT FROM AUTHOR]