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Solving the Einstein constraints on multi-block triangulations using finite elements.
by Oleg Korobkin, Burak Aksoylu, Michael Holst, Enrique Pazos, Manuel Tiglio
Abstract:
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 $\psi$. 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.
UCSD-CCoM-08-02.pdf January 2008
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Randolph E. Bank
Philip E. Gill
Michael Holst
Administrative Contact:
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by Oleg Korobkin, Burak Aksoylu, Michael Holst, Enrique Pazos, Manuel Tiglio
Abstract:
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 $\psi$. 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.
UCSD-CCoM-08-02.pdf January 2008