We study the fully explicit numerical approximation of a semilinear elliptic boundary value model problem, which features a monomial reaction and analytic forcing, in a bounded polygon $\Omega\subset\mathbb{R}^2$ with a finite number of straight edges. In particular, we analyze the convergence of $hp$-type iterative linearized Galerkin ($hp$-ILG) solvers. Our convergence analysis is carried out for conforming $hp$-finite element (FE) Galerkin discretizations on sequences of regular, simplicial partitions of $\Omega$, with geometric corner refinement, with polynomial degrees increasing in sync with the geometric mesh refinement towards the corners of $\Omega$. For a sequence of discrete solutions generated by the ILG solver, with a stopping criterion that is consistent with the exponential convergence of the exact $hp$-FE Galerkin solution, we prove exponential convergence in $\mathrm{H}^1(\Omega)$ to the unique weak solution of the boundary value problem. Numerical experiments illustrate the exponential convergence of the numerical approximations obtained from the proposed scheme in terms of the number of degrees of freedom as well as of the computational complexity involved.

翻译：本文研究了定义在具有有限条直边的有界多边形域$\Omega\subset\mathbb{R}^2$上，具有单项反应与解析强迫项的半线性椭圆边值问题的全显式数值逼近。特别地，我们分析了$hp$型迭代线性化伽辽金($hp$-ILG)求解器的收敛性。收敛性分析针对$\Omega$上正则单纯形剖分序列（带几何角部加密）以及随角部网格几何加密而同步增加的多项式次数，在协调$hp$有限元伽辽金离散框架下进行。对于ILG求解器生成的离散解序列，当采用与精确$hp$-FE伽辽金解指数收敛性一致的停止准则时，我们证明其在$\mathrm{H}^1(\Omega)$空间中指数收敛到边值问题的唯一弱解。数值实验通过自由度数量和计算复杂度两方面，展示了所提格式获得的数值逼近的指数收敛性。