论文标题
与椭圆形和抛物线问题的不连续的Galerkin近似与Dirac Line源
Discontinuous Galerkin Approximations to Elliptic and Parabolic Problems with a Dirac Line Source
论文作者
论文摘要
介绍了任何订单K的内部惩罚不连续的Galerkin方法的分析，用于解决dirac Line源的椭圆形和抛物线问题。对于稳态情况，我们通过在L2规范和加权能量规范中得出先验误差估计来证明该方法的收敛性。此外，对于任何近似顺序，我们证明了能量规范中几乎最佳的局部误差估计。此外，对于分段线性近似的情况，获得了L2规范中几乎最佳的局部误差估计值，而L2 Norm中的次优误差界显示了任何多项式程度。对于时间依赖性情况，通过证明在时间和空间中的L2中证明误差估计值来确定半污染和向后欧拉完全离散方案的收敛性。添加椭圆问题的数值结果以支持理论结果。
The analyses of interior penalty discontinuous Galerkin methods of any order k for solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence of the method by deriving a priori error estimates in the L2 norm and in weighted energy norms. In addition, we prove almost optimal local error estimates in the energy norm for any approximation order. Further, almost optimal local error estimates in the L2 norm are obtained for the case of piecewise linear approximations whereas suboptimal error bounds in the L2 norm are shown for any polynomial degree. For the time-dependent case, convergence of semi-discrete and of backward Euler fully discrete scheme is established by proving error estimates in L2 in time and in space. Numerical results for the elliptic problem are added to support the theoretical results.
