平稳二阶平均场博弈的有限元逼近的近似与完全拟最优性
Near and full quasi-optimality of finite element approximations of stationary second-order mean field games
AI总结：
本文针对平稳二阶平均场博弈的有限元离散，证明了误差的近似拟最优性，并在严格锐角网格及凸域情形下分别获得完全拟最优性和最优收敛阶。
摘要：
我们在Lipschitz多面体域上建立了单调稳定化有限元离散化平稳二阶平均场博弈（MFG）的先验误差界。在适当假设下，我们证明了该逼近在$H^1$范数意义下渐近地接近拟最优，即在足够细的网格上，精确解与计算解之间的误差由相应有限元空间的最佳逼近误差加上可能由于稳定化而产生的额外项所界定，该额外项相对于网格尺寸具有最优阶。由此，对于具有足够正则性的解，我们推导出误差关于网格尺寸的最优收敛阶。我们进一步在严格锐角网格序列的更受限情形下证明了逼近误差的完全渐近拟最优性。我们的第三个主要贡献是，在区域为凸的情况下，即使密度函数仅具有$H^1$中的最小正则性，值函数逼近的$H^1$范数误差的收敛阶仍然保持最优。
英文摘要：
We establish a priori error bounds for monotone stabilized finite element discretizations of stationary second-order mean field games (MFG) on Lipschitz polytopal domains. Under suitable hypotheses, we prove that the approximation is asymptotically nearly quasi-optimal in the $H^1$-norm in the sense that, on sufficiently fine meshes, the error between exact and computed solutions is bounded by the best approximation error of the corresponding finite element space, plus possibly an additional term, due to the stabilization, that is of optimal order with respect to the mesh size. We thereby deduce optimal rates of convergence of the error with respect to the mesh-size for solutions with sufficient regularity. We further show full asymptotic quasi-optimality of the approximation error in the more restricted case of sequences of strictly acute meshes. Our third main contribution is to further show, in the case where the domain is convex, that the convergence rate for the $H^1$-norm error of the value function approximation remains optimal even if the density function only has minimal regularity in $H^1$.
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