Error Analysis of an Unfitted HDG Method for a Class of Non-linear Elliptic Problems
Error Analysis of an Unfitted HDG Method for a Class of Non-linear Elliptic Problems
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DOI：
10.1007/s10915-022-01767-1
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发表时间：
2021-05
影响因子：
2.5
通讯作者：
N'estor S'anchez;Tonatiuh S'anchez-Vizuet;Manuel E. Solano
中科院分区：
文献类型：
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作者：
N'estor S'anchez;Tonatiuh S'anchez-Vizuet;Manuel E. Solano
研究了一类源项和扩散系数均为非线性的曲线域非线性内椭圆型边值问题的可杂交间断伽辽金离散化。我们考虑非线性扩散系数取决于解和解的梯度的情况。为了避免对曲面元素的需要，离散解在多边形子域上计算，该子域不假定插值到真实边界，从而产生未拟合的计算网格。我们证明，在对源项和计算域的温和假设下，离散系统是定态良好的。此外，我们提供了一个先验误差估计，表明只要弯曲边界和计算边界之间的距离保持与网格参数相同的数量级，离散解将具有最优收敛阶。
We study Hibridizable Discontinuous Galerkin (HDG) discretizations for a class of non-linear interior elliptic boundary value problems posed in curved domains where both the source term and the diffusion coefficient are non-linear. We consider the cases where the non-linear diffusion coefficient depends on the solution and on the gradient of the solution. To sidestep the need for curved elements, the discrete solution is computed on a polygonal subdomain that is not assumed to interpolate the true boundary, giving rise to an unfitted computational mesh. We show that, under mild assumptions on the source term and the computational domain, the discrete systems are well posed. Furthermore, we provide a priori error estimates showing that the discrete solution will have optimal order of convergence as long as the distance between the curved boundary and the computational boundary remains of the same order of magnitude as the mesh parameter.