AI 中文总结
该研究针对混合有限元方法中狄拉克源导致的对偶性失效问题，提出散度形式分裂技术，推导了通量的L^p误差阶，通过网格分级恢复计算复杂度，得到可靠的L^p残差估计量。
AI 中文摘要
对于混合有限元方法，狄拉克源首先会导致对偶性失效，而非正则性失效：守恒方程需在勒贝格空间上进行检验，而狄拉克测度不属于任何此类空间的对偶空间。因此，我们通过散度形式分裂将测度从守恒律中移除：从物理通量中减去一个散度为狄拉克测度的显式场，将修正后的通量作为混合未知量，使守恒方程中仅保留载荷的正则部分。等价地，且与任何离散化无关，狄拉克问题可重写为一个椭圆方程，其数据为散度形式，由L^p场生成。该减去的场仅取决于源的位置，与系数无关。无需依赖系数的奇异解，也无需离散δ函数，仅RT_0-P_0系统的载荷向量发生变化，源可相对于网格位于任意位置：顶点处、系数界面上或单元内部。除非在源处将分裂与算子匹配，否则修正后的通量属于所有p<2的L^p空间，但不属于L^2空间，因此通量误差分析需脱离希尔伯特尺度。我们证明了通量的准最佳逼近界，并由此得出，在拟一致网格族上，通量误差的阶恰好为h^(2/p-1)：匹配的下界来自承载源的单个单元。在该位置对网格分级可恢复一阶复杂度，即单元数N的-1/2次方，自适应计算可达到该精度。标量变量仅受限于解的分段常数逼近，而它恰好能达到该逼近精度。我们还证明了勒贝格尺度下的残差范数等价性，从而得到一个可计算的L^p估计量，对混合通量及恢复的势而言，该估计量可靠且局部有效。
英文摘要
For a mixed finite element method, a Dirac source is first a failure of duality, not of regularity: the conservation equation is tested against a Lebesgue space, and a Dirac measure lies in the dual of none. We therefore remove the measure from the conservation law by a divergence-form splitting. An explicit field whose divergence is the Dirac measure is subtracted from the physical flux, and the modified flux is taken as the mixed unknown, so that only the regular part of the load remains in the conservation equation. Equivalently, and independently of any discretization, the Dirac problem is rewritten as an elliptic equation whose data are in divergence form, generated by a field of L^p. The subtracted field depends on the location of the source alone and not on the coefficient. No coefficient-dependent singular solution and no discrete delta is needed, only the load vector of the RT_0-P_0 system changes, and the source may sit anywhere relative to the mesh: at a vertex, on a coefficient interface, or inside an element. Unless the splitting is matched to the operator at the source, the modified flux lies in L^p for every p<2 but not in L^2, so the flux error analysis has to leave the Hilbert scale. We prove a quasi-best approximation bound for the flux, and with it that on a quasi-uniform family the flux error is exactly of order h^(2/p-1): the matching lower bound comes already from the single element carrying the source. Grading the mesh there restores first-order complexity, N^(-1/2) in the number of elements, and the adaptive computations attain it. The scalar variable is limited only by piecewise constant approximation of the solution, which it attains. We also prove a residual norm equivalence in the Lebesgue scale, yielding a computable L^p estimator, reliable and locally efficient for the mixed flux together with a recovered potential.
