AI 中文总结
针对 Grad--Mercier 方程提出协调有限元方法，通过截断秩定律和节点质量集中处理平台，得到唯一离散平衡态，并证明收敛性。
AI 中文摘要
我们针对 Grad--Mercier 方程（一种磁平衡的非局部椭圆模型，其源项依赖于通量函数的全局秩）开发并分析了一种协调的分片仿射有限元方法。主要困难在于，唯一的连续平衡态具有一个具有指定正面积的极大平台。因此，解仅在平台质量之上采样秩定律，而分离相等值的扰动可能将该质量分布在多个数值秩上，从而改变非严格的秩源。我们利用连续问题的适定性结构，将秩定律替换为等价的截断非负定律，并将 Lebesgue 测度替换为质量集中节点测度。相等的节点值被赋予一个共同的秩区间，从而保留每个平层的全部集中质量。由此产生的加权重排给出严格凸的离散能量，从而得到唯一的有限元平衡态。在满足离散比较原理的网格上，离散平衡态是非负的，并且以可计算的障碍函数为上界。一种泊松预条件重排迭代在每个固定网格上收敛，并允许计算后验能量和能量范数界。当多边形有限元域收敛到物理截面时，离散平衡态在 $H^1_0(\Omega)$ 中强收敛，并在 $\overline\Omega$ 上一致收敛到唯一的 Grad--Mercier 平衡态。重建的平衡源在 $L^{\infty}(\Omega)$ 中弱-* 收敛到原始非局部源。
英文摘要
We develop and analyse a conforming piecewise affine finite element method for the Grad--Mercier equation, a nonlocal elliptic model for magnetic equilibria whose source depends on the global rank of a flux function. The principal difficulty is that the unique continuum equilibrium has a maximum plateau of prescribed positive area. The solution therefore samples the rank law only above the plateau mass, while perturbations that separate equal values may distribute this mass over several numerical ranks and thereby alter the non-strict rank source. We exploit the continuum well-posedness structure to replace the rank law by an equivalent clipped, nonnegative law, and replace Lebesgue measure by a mass-lumped nodal measure. Equal nodal values are assigned a common rank interval, thereby retaining the full lumped mass of each tied level. The resulting weighted rearrangement gives a strictly convex discrete energy and hence a unique finite element equilibrium. On meshes satisfying a discrete comparison principle, the discrete equilibrium is nonnegative and bounded above by a computable barrier. A Poisson-preconditioned rearrangement iteration converges on each fixed mesh and admits computable a posteriori energy and energy-norm bounds. As polygonal finite element domains converge to the physical cross-section, the discrete equilibria converge strongly in $H^1_0(Ω)$ and uniformly on $\overlineΩ$ to the unique Grad--Mercier equilibrium. Reconstructed equilibrium sources converge weak-* in $L^{\infty}(Ω)$ to the original nonlocal source.
Comments24 pages, 3 figures, 2 tables
