用于Signorini问题的障碍正则化对称Nitsche方法
A Barrier-Regularized Symmetric Nitsche Method for the Signorini Problem
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中文总结 AI 辅助
该研究提出并分析了用于Signorini问题的障碍正则化对称Nitsche方法，证明其离散问题的可解性、收敛速率等理论结果，数值实验验证了相关理论并发现平滑处理难以良好分辨接触集。
中文摘要 AI 辅助
我们引入并分析了一种用于标量Signorini问题的障碍正则化对称Nitsche方法。在增广拉格朗日方法中对非负松弛变量应用对数障碍，随后消除该变量，得到了光滑的正部算子以及原对偶中心路径上的互补关系，其中障碍参数为μ=γs。对于任意s>0，当Nitsche参数足够大时，离散问题是连续可微的、唯一可解的，且具有对称正定的牛顿矩阵。在温和的障碍可行性条件下，对于任意μ>0，连续对数能量具有唯一的极小值u_μ，其间隙几乎处处为正，且||u-u_μ||_{H^1(Ω)}≲μ^{1/2}。在有限元近似所用的Sobolev正则性下，该极小值满足L^2中心路径边界定律。若障碍局部为H^2函数的迹，我们还证明了在每个平面接触面内部，于可能包含极限Signorini解自由边界点的邻域上，存在一致局部H^2界。该方法相对于u_μ是完全相容且拟最优的，常数与s无关。若u_μ在H^r(Ω)中一致有界，其中3/2<r≤k+1，则||u-u_h||_{H^1(Ω)}≲h^{r-1}||u_μ||_{H^r(Ω)}+μ^{1/2}；因此，充分平衡s≲h^{2r-1}可保持可用的能量范数收敛速率。我们还推导了离散穿透和互补残差的收敛速率。在结构化和非结构化网格上使用P1和P2单元进行的数值实验验证了预测的收敛速率和局部牛顿理论，同时表明保持速率的平滑处理仍可能无法良好分辨接触集。
英文摘要
We introduce and analyze a barrier-regularized symmetric Nitsche method for the scalar Signorini problem. Applying a logarithmic barrier to the nonnegative slack variable in an augmented Lagrangian and then eliminating that variable yields a smooth positive-part operator and the perturbed complementarity relation on a primal-dual central path, with barrier parameter $μ=γs$. For every $s>0$, the discrete problem is continuously differentiable, uniquely solvable, and has a symmetric positive definite Newton matrix when the Nitsche parameter is sufficiently large. Under a mild barrier-feasibility condition, the continuous logarithmic energy has a unique minimizer $u_μ$ for every $μ>0$, its gap is positive almost everywhere, and $\|u-u_μ\|_{H^1(Ω)}\lesssimμ^{1/2}$. Under the Sobolev regularity used for finite element approximation, this minimizer satisfies the \(L^2\) central-path boundary law. If the obstacle is locally the trace of an $H^2$-function, we also prove a uniform local $H^2$ bound in the interior of each planar contact face, on neighborhoods that may contain a free-boundary point of the limiting Signorini solution. The method is exactly consistent and quasi-optimal relative to $u_μ$, with constants independent of $s$. If $u_μ$ is uniformly bounded in $H^r(Ω)$, $3/2<r\leq k+1$, then $\|u-u_h\|_{H^1(Ω)}\lesssim h^{r-1}\|u_μ\|_{H^r(Ω)}+μ^{1/2}$; hence the sufficient balance $s\lesssim h^{2r-1}$ preserves the available energy-norm rate. We also derive rates for discrete penetration and the complementarity residual. Numerical experiments with $P_1$ and $P_2$ elements on structured and unstructured meshes support the predicted rates and the local Newton theory, while showing that a rate-preserving smoothing may still resolve the contact set poorly.
