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In Klingenberg, Schnücke and Xia (Math. Comp. 86 (2017), 1203-1232) an arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) method to solve conservation laws has been developed and analyzed. In this paper, the ALE-DG method will be extended to several dimensions. The method will be designed for simplex meshes. This will ensure that the method satisfies the geometric conservation law, if the accuracy of the time integrator is not less than the value of the spatial dimension. For the semi-discrete method the L 2 -stability will be proven. Furthermore, an error estimate which provides the suboptimal (k + 1 2 ) convergence with respect to the L ∞ 0, T ; L 2 (Ω) -norm will be presented, when an arbitrary monotone flux is used and for each cell the approximating functions are given by polynomials of degree k. The two dimensional fully-discrete explicit method will be combined with the bound preserving limiter developed by Zhang, Xia and Shu in (J. Sci. Comput. 50 (2012), 29-62). This limiter does not affect the high order accuracy of a numerical method. Then, for the ALE-DG method revised by the limiter the validity of a discrete maximum principle will be proven. The numerical stability, robustness and accuracy of the method will be shown by a variety of two dimensional computational experiments on moving triangularThe numerical flux function needs to satisfy certain properties. These properties are discussed in the Section 2.4.Finally, on the reference cell, the semi-discrete ALE-DG method appears as the following problem:Problem 1 (The semi-discrete ALE-DG method on the reference cell). Find a function(4.5)Proof. Since f (c) contains merely constant coefficients andthe integration by parts formula providesThus, we obtain the identity (4.5) by (4.6) and (4.7).Next, we assume that u * h = c solves the semi-discrete ALE-DG method Problem 1. Then, we obtain by (4.4) and (4.5)(4.8) 17The equation (4.8) and the ODE (2.12) are equivalent, since c is an arbitrary constant,is an arbitrary test function and the quantitiesare merely time-dependent. We note that the time evolution of the metric terms J K(t) needs to be respected in the time discretization of the semi-discrete ALE-DG method Problem 1. Therefore, we discretize the ODE (2.12) and (4.4) simultaneously by the same TVD-RK method. The stage solutions of the TVD-RK discretization for (2.12) will be used to update the metric terms in the TVD-RK discretization for (4.4).The fully-discrete ALE-DG method: First, the ODE (2.12) is discretized by a s-stage TVD-RK method:where K n+γ j := K (t n + γ j t) and ω n+γ j := ω (t n + γ j t). The stage solutions {J K n,i } s i=0 are used to update the metric terms in the TVD-RK discretization of (4.4). The Runge-Kutta method needs to solve the ODE (2.12) exact such thatwhere T (t n+1 ) is the regular mesh of simplices which has been used in the Section 2.1 to construct the time-dependent cells (2.3). We note that in a d-dimensional space a TVD-RK method with order greater than or equal to d is necessary to compute the metric ...

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Finite difference and pseudo-spectral methods 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 less frequently 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 aiming 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 partly the reason that the accuracy of finite element method is higher than the finite difference method. At the same time, different kinds of formulation parameters setting are also discussed.

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