Well-posedness of hp-version discontinuous Galerkin methods for fractional diffusion wave equations
Well-posedness of hp-version discontinuous Galerkin methods for fractional diffusion wave equations
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DOI：
10.1093/imanum/drt048
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发表时间：
2014-10
影响因子：
2.1
通讯作者：
K. Mustapha;D. Schötzau
中科院分区：
文献类型：
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作者：
K. Mustapha;D. Schötzau
建立了求解分数阶超扩散演化问题的hp型时步间断Galerkin方法的适定性。特别地，我们证明了具有非均匀时间步长和可变逼近度的一般hp型有限元空间近似解的存在性和唯一性。然后，我们得到新的HP版本的误差估计在非标准的范数，这是完全明确的局部离散化和正则性参数。因此，我们表明，通过采用几何细化的时间步长和线性增加的逼近阶数，时间自由度的数量的指数收敛率实现的解决方案与奇异（时间）的行为在t=0附近所造成的弱奇异内核。此外，我们证明了最佳的代数收敛速度的h-版本的近似分级网格。我们提出了一系列的数值试验，我们验证实验，我们的理论收敛性也成立，在较强的L∞范数。
We establish the well-posedness of an hp-version time-stepping discontinuous Galerkin method for the numerical solution of fractional superdiffusion evolution problems. In particular, we prove the existence and uniqueness of approximate solutions for generic hp-version finite element spaces featuring nonuniform time steps and variable approximation degrees. We then derive new hp-version error estimates in a nonstandard norm, which are completely explicit in the local discretization and regularity parameters. As a consequence, we show that by employing geometrically refined time steps and linearly increasing approximation orders, exponential rates of convergence in the number of temporal degrees of freedom are achieved for solutions with singular (temporal) behaviour near t=0 caused by the weakly singular kernel. Moreover, we show optimal algebraic convergence rates for h-version approximations on graded meshes. We present a series of numerical tests where we verify experimentally that our theoretical convergence properties also hold true in the stronger L∞norm.