  Finite difference method and pseudo-spectral method have been widely used in
the numerical relativity to solve the Einstein equations. As the third major
category method to solve partial differential equations, finite element method
is much less used in numerical relativity. In this paper we design a finite
element algorithm to solve the evolution part of the Einstein equations. This
paper is the second one of a systematic investigation of applying adaptive
finite element method to the Einstein equations, especially aim for binary
compact objects simulations. The first paper of this series has been
contributed to the constrained part of the Einstein equations for initial data.
Since applying finite element method to the Einstein equations is a big
project, we mainly propose the theoretical framework of a finite element
algorithm together with local discontinuous Galerkin method for the Einstein
equations in the current work. In addition, we have tested our algorithm based
on the spherical symmetric spacetime evolution. In order to simplify our
numerical tests, we have reduced the problem to a one-dimensional space problem
by taking the advantage of the spherical symmetry. Our reduced equation system
is a new formalism for spherical symmetric spacetime simulation. Based on our
test results, we find that our finite element method can capture the shock
formation which is introduced by numerical error. In contrast, such shock is
smoothed out by numerical dissipation within the finite difference method. We
suspect this is the part reason for that the accuracy of finite element method
is higher than finite difference method. At the same time kinds of formulation
parameters setting are also discussed.
