Numerical approximation of bi-harmonic wave maps into spheres

Ľubomír Baňas, Sebastian Herr
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Abstract

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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进入球面的双谐波图的数值逼近
我们为双谐波映射到球面方程构建了一个结构保留的非符合有限元逼近方案。它满足离散能量定律,并保留了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法器在主题点强制执行。这种方法将空间近似限制在(不一致的)线性有限元上。我们证明,数值近似在空间维数 $d=1$ 时收敛于连续问题的弱解。由于缺乏离散乘积规则,以及数值近似在非构造集合中的低正则性,在维数 $d>1$ 下的收敛分析变得复杂。因此,我们通过在数值近似中引入额外的稳定项来证明数值近似在更高维度上的收敛性。我们通过数值实验证明了所提出的数值近似的性能,并说明了双拉普拉奇的奇异效果,它可以防止奇异现象的形成。
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