双调和波映射到球面的数值近似
Numerical approximation of bi-harmonic wave maps into spheres
AI总结：
本文提出一种保持结构的非匹配有限元方法，用于双调和波映射到球面方程的数值近似，该方法满足离散能量守恒并保持连续问题的非凸球面约束，通过离散拉格朗日乘数在网格点上强制约束，且在高维情况下引入额外稳定化项以保证收敛性。
摘要：
我们构造了一种保持结构的非匹配有限元近似方案，用于双调和波映射到球面方程。该方法满足离散能量守恒并保持连续问题的非凸球面约束。离散球面约束通过离散拉格朗日乘数在网格点上强制。该方法将空间近似限制为（非匹配）线性有限元。我们证明在空间维数d=1时，数值近似收敛到连续问题的弱解。在d>1的维度中，收敛分析复杂，因为缺乏离散乘积法则以及非匹配设置中数值近似的低正则性。因此，我们通过在数值近似中引入额外的稳定化项来展示在高维情况下的收敛性。我们通过数值实验展示所提出数值近似的性能，并说明双调和算子的正则化效应，防止奇异性的形成。
英文摘要：
We construct a structure preserving non-conforming finite element approximation scheme for the bi-harmonic wave maps into spheres equation. It satisfies a discrete energy law and preserves the non-convex sphere constraint of the continuous problem. The discrete sphere constraint is enforced at the mesh-points via a discrete Lagrange multiplier. This approach restricts the spatial approximation to the (non-conforming) linear finite elements. We show that the numerical approximation converges to the weak solution of the continuous problem in spatial dimension $d=1$. The convergence analysis in dimensions $d>1$ is complicated by the lack of a discrete product rule as well as the low regularity of the numerical approximation in the non-conforming setting. Hence, we show convergence of the numerical approximation in higher-dimensions by introducing additional stabilization terms in the numerical approximation. We present numerical experiments to demonstrate the performance of the proposed numerical approximation and to illustrate the regularizing effect of the bi-Laplacian which prevents the formation of singularities.
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