多块三角剖分上Einstein约束方程的有限元求解
Solving the Einstein constraint equations on multi-block triangulations using finite element methods
摘要：
为了生成用于非线性相对论模拟的初始数据，需要求解爱因斯坦约束，这可以转换为一组耦合的非线性椭圆方程。在这里，我们提出了一种方法来解决这些方程的三维多块域使用有限元方法。我们说明了我们的方法上的一个简单的例子Brill波的初始数据，约束减少到一个单一的线性椭圆方程的共形因子$ψ$。我们使用二次拉格朗日元素的半结构化单纯网格，通过三角形的多块网格。在均匀细化的情况下，该计划是超收敛的网格顶点，由于局部对称的有限元基础上的局部空间反演。我们表明，在超收敛的情况下，随后的非结构化网格细化不提高我们的初始数据的质量。作为概念的证明，这种方法是可行的，在三维中产生多块初始数据，构建的初始数据后，我们演变他们在时间上使用高阶有限差分多块方法和提取的引力波的数值解。
英文摘要：
In order to generate initial data for nonlinear relativistic simulations, one needs to solve the Einstein constraints, which can be cast into a coupled set of nonlinear elliptic equations. Here we present an approach for solving these equations on three-dimensional multi-block domains using finite element methods. We illustrate our approach on a simple example of Brill wave initial data, with the constraints reducing to a single linear elliptic equation for the conformal factor $ψ$. We use quadratic Lagrange elements on semi-structured simplicial meshes, obtained by triangulation of multi-block grids. In the case of uniform refinement the scheme is superconvergent at most mesh vertices, due to local symmetry of the finite element basis with respect to local spatial inversions. We show that in the superconvergent case subsequent unstructured mesh refinements do not improve the quality of our initial data. As proof of concept that this approach is feasible for generating multi-block initial data in three dimensions, after constructing the initial data we evolve them in time using a high order finite-differencing multi-block approach and extract the gravitational waves from the numerical solution.
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