An h-p Version of the Continuous Petrov-Galerkin Finite Element Method for Volterra Integro-Differential Equations with Smooth and Nonsmooth Kernels
An h-p Version of the Continuous Petrov-Galerkin Finite Element Method for Volterra Integro-Differential Equations with Smooth and Nonsmooth Kernels
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DOI：
10.1137/15m1006489
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发表时间：
2015-12
期刊：
影响因子：
--
通讯作者：
Lijun Yi;B. Guo
中科院分区：
文献类型：
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作者：
Lijun Yi;B. Guo
我们提出了连续 Petrov-Galerkin (CPG) 有限元方法的 $h$-$p$ 版本，用于具有光滑和非光滑核的线性 Volterra 积分微分方程。我们在 $L^2$-、$H^1$- 和 $L^\infty$-范数中建立先验误差估计，这些范数对于局部离散化和正则性参数是完全明确的。对于由弱奇异核引起的奇异解，我们证明了具有准均匀网格的 CPG 方法的 $h$-$p$ 版本可以实现代数收敛速度。此外，我们表明，通过使用几何时间分区和线性增加多项式次数，可以实现具有启动奇点的解的指数收敛率。提供数值实验来说明理论结果。
We present an $h$-$p$ version of the continuous Petrov--Galerkin (CPG) finite element method for linear Volterra integro-differential equations with smooth and nonsmooth kernels. We establish a priori error estimates in the $L^2$-, $H^1$-, and $L^\infty$-norms that are completely explicit with respect to the local discretization and regularity parameters. For singular solutions caused by the weakly singular kernels, we prove that algebraic convergence rates can be achieved for the $h$-$p$ version of the CPG method with quasi-uniform meshes. Moreover, we show that exponential rates of convergence can be achieved for solutions with start-up singularities by using geometric time partitions and linearly increasing polynomial degrees. Numerical experiments are provided to illustrate the theoretical results.