论文标题
使用张量产品有限元素扩展fenics在更高维度上工作
Extending FEniCS to Work in Higher Dimensions Using Tensor Product Finite Elements
论文作者
论文摘要
我们提出了一种扩展有限元库fenics的方法，以通过构建张量产品有限元元素来解决以上域以上域中的域问题。该方法仅要求高维域的结构为两个较低维子域的笛卡尔产物。在这项研究中，我们考虑了标量线性偏微分方程的Dirichlet问题，尽管该方法可以扩展到非线性问题。张量产品有限元的利用使我们能够构建一个线性代数方程的全局系统，该系统仅依赖于fenics中包含的下维子域的有限元基础架构。我们在四个独特的测试案例中证明了我们方法的有效性。第一个测试用例是在四维域中提出的泊松方程，该方程是使用经典的Galerkin有限元方法求解的两个单位正方形的笛卡尔产物。第二个测试用例是时空中的波方程，其中计算域是二维空间网格和一个维度时间间隔的笛卡尔产物。在第二种情况下，我们还采用了Galerkin方法。第三个测试案例是一个以对流为主的对流扩散方程，其中全局域是两个一个维度间隔的笛卡尔产物，在该范围内，应用了流线上的彼得罗夫 - 加密金方法来确保离散的稳定性。最终的测试用例使用Galerkin方法在两个间隔的笛卡尔产物上解决泊松问题，其空间变化，不可分割的扩散术语。在所有情况下，使用p = 1基础，并且相对于H细化，均可达到误差的最佳l^2收敛率。
We present a method to extend the finite element library FEniCS to solve problems with domains in dimensions above three by constructing tensor product finite elements. This methodology only requires that the high dimensional domain is structured as a Cartesian product of two lower dimensional subdomains. In this study we consider Dirichlet problems for scalar linear partial differential equations, though the methodology can be extended to non-linear problems. The utilization of tensor product finite elements allows us to construct a global system of linear algebraic equations that only relies on the finite element infrastructure of the lower dimensional subdomains contained in FEniCS. We demonstrate the effectiveness of our methodology in four distinctive test cases. The first test case is a Poisson equation posed in a four dimensional domain which is a Cartesian product of two unit squares solved using the classical Galerkin finite element method. The second test case is the wave equation in space-time, where the computational domain is a Cartesian product of a two dimensional space grid and a one dimensional time interval. In this second case we also employ the Galerkin method. The third test case is an advection dominated advection-diffusion equation where the global domain is a Cartesian product of two one dimensional intervals in which the streamline upwind Petrov-Galerkin method is applied to ensure discrete stability. The final test case uses the Galerkin approach to solve a Poisson problem on a Cartesian product of two intervals with a spatially varying, non-separable diffusivity term. In all cases, a p=1 basis is used and optimal L^2 convergence rates of order h^{p+1} of the errors are achieved with respect to h refinement
