In this paper we present a minimax projection method for linear evolution equations in Hilbert space. The method extends classical Galerkin approach: it builds a differential-algebraic equation with uncertain parameters that models dynamics of exact projection coefficients representing the projection of the evolution equation's solution onto a finitedimensional subspace. The a priori ellipsoidal bounding set for uncertain parameters is also constructed. The output of the method is an ellipsoid enclosing exact projection coefficients. The ellipsoid can be constructed numerically: we illustrate this applying the method to 1D heat equation. ©2013 IEEE.