We combine the newly-constructed Galerkin difference basis with the energy-based discontinuous Galerkin method for wave equations in second order form. The approximation properties of the resulting method are excellent and the allowable time steps are large compared to traditional discontinuous Galerkin methods. The one drawback of the combined approach is the cost of inversion of the local mass matrix. We demonstrate that for constant coefficient problems on Cartesian meshes this bottleneck can be removed by the use of a modified Galerkin difference basis. For variable coefficients or non-Cartesian meshes this technique is not possible and we instead use the preconditioned conjugate gradient method to iteratively invert the mass matrices. With a careful choice of preconditioner we can demonstrate optimal complexity, albeit with a larger constant.
翻译:我们把新构建的Galerkin差异基础与以能量为基础的不连续 Galerkin 方法相结合,以第二顺序形式处理波形方程式。 所产生的方法的近似特性极佳, 与传统的不连续 Galerkin 方法相比, 允许的时间步骤大。 合并方法的一个缺点是当地质量矩阵的反向成本。 我们证明, 对于Cartesian meshes 的恒定系数问题, 使用修改的Galerkin 差异基础可以消除这一瓶颈。 对于可变系数或非Carterkin meshes 来说, 这种方法是不可能的, 而我们则使用预设的同位梯度方法来反复反转质量矩阵。 谨慎地选择先决条件, 我们可以展示最佳的复杂度, 尽管使用更大的常数 。