Discontinuous Galerkin approximations to elliptic and parabolic problems with a Dirac line source
Discontinuous Galerkin approximations to elliptic and parabolic problems with a Dirac line source
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狄拉克线源椭圆和抛物线问题的不连续伽辽金逼近
DOI：
10.1051/m2an/2022095
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发表时间：
2023
期刊：
影响因子：
--
通讯作者：
Riviere, Beatrice
中科院分区：
文献类型：
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作者：
Masri, Rami;Shen, Boqian;Riviere, Beatrice
给出了求解具有Dirac线源的椭圆型和抛物型问题的任意阶内罚不连续伽辽金方法的分析。对于稳态情况，我们通过推导2范数和加权能量范数下的优先误差估计来证明该方法的收敛性。此外，我们还证明了对任意近似阶的能量范数的几乎最优局部误差估计。此外，对于分段线性近似的情况，得到了thel2范数中几乎最优的局部误差估计，而对于任何多项式次，thel2范数中的次优误差界都显示出来。对于与时间相关的情况，通过证明在时间和空间上的误差估计，证明了半离散格式和后向欧拉完全离散格式的收敛性。文中还加入了椭圆型问题的数值结果来支持理论结果。
The analyses of interior penalty discontinuous Galerkin methods of any orderkfor solving elliptic and parabolic problems with Dirac line sources are presented. For the steady state case, we prove convergence of the method by derivinga priorierror estimates in theL2norm 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 theL2norm are obtained for the case of piecewise linear approximations whereas suboptimal error bounds in theL2norm 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 inL2in time and in space. Numerical results for the elliptic problem are added to support the theoretical results.