We establish estimates on the error made by the Ritz method for quadratic energies on the space $H^1(\Omega)$ in the approximation of the solution of variational problems with different boundary conditions. Special attention is paid to the case of Dirichlet boundary values which are treated with the boundary penalty method. We consider arbitrary and in general non linear classes $V\subseteq H^1(\Omega)$ of ansatz functions and estimate the error in dependence of the optimisation accuracy, the approximation capabilities of the ansatz class and - in the case of Dirichlet boundary values - the penalisation strength $\lambda$. For non-essential boundary conditions the error of the Ritz method decays with the same rate as the approximation rate of the ansatz classes. For the boundary penalty method we obtain that given an approximation rate of $r$ in $H^1(\Omega)$ and an approximation rate of $s$ in $L^2(\partial\Omega)$ of the ansatz classes, the optimal decay rate of the estimated error is $\min(s/2, r) \in [r/2, r]$ and achieved by choosing $\lambda_n\sim n^{s}$. We discuss how this rate can be improved, the relation to existing estimates for finite element functions as well as the implications for ansatz classes which are given through ReLU networks. Finally, we use the notion of $\Gamma$-convergence to show that the Ritz method converges for a wide class of energies including nonlinear stationary PDEs like the $p$-Laplace.
翻译:我们设定了Ritz 方法对空间 $H1 (\\ OMEGA) 的二次能量的误差的估算值。 对于不同边界条件的变异性问题的近似解决办法,我们特别注意Drichlet 边界值的误差,这些值与边界处罚方法的近似率相同。 我们考虑asatz 函数的任意和一般非线性等级 $V\ subseteq H1 (\ OMEGAGA), 并估计了对优化精确度的依赖性差错、 ansatz 类的近似能力, 以及 - 在 Dirichlet 边界值的近似值中 -- 惩罚强度 $\ lambda 。 对于非必要的边界条件, Ritz 方法的误差差率与asatz 等级的近似率相同。 对于axx $1 (\ Omega) 和 $ an slax lax lax lax le leg le le lex lex level ections r= r\\\\\\\\ r} we lex a lex lax lax lax lax lax lax lex lex lex, lex lex lex lex a lex lex lex lex lex lex lex lex lex r= r= r/2,我们 r= lex r= r/2,我们 r= r=xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx