Shun Zhang

Research Interests 研究興趣

My research focuses on finite element methods, particularly least-squares and mixed formulations, adaptive algorithms, singular-source and interface problems, reduced-order modeling, and scientific machine learning.

我的研究聚焦於有限元方法，尤其是最小二乘與混合格式、自適應算法、奇異源與界面問題、降階建模，以及科學機器學習。

Featured Research 代表性研究

• Least-squares finite element methods 最小二乘有限元方法
  LSFEM coercivity Where Does LSFEM Coercivity Come From? PDE stability yields LSFEM coercivity; inf-sup is its linear variational expression. Nonconforming methods require their own stability analysis. LSFEM 的強制性究竟從何而來？ LSFEM coercivity 來自 PDE stability；inf-sup 是線性問題的變分表述。Nonconforming 方法則需另外分析穩定性。

  LSFEM with weakly imposed boundary conditions How Weak Can Boundary Enforcement Be? Sharp boundary weights: h_F⁻¹ for Dirichlet, 1 for Neumann. PDE stability yields norm and error equivalences, with fully discontinuous extensions. 弱施加邊界條件，權重究竟可以多弱？ Dirichlet 使用 h_F⁻¹，Neumann 使用 1；由 PDE stability 得到範數與誤差等價，並延伸到 fully discontinuous 離散。

  H⁻¹ data and first-order system least squares Do We Really Need an H⁻¹-Norm LSFEM? A divergence-form splitting and the correct boundary treatment retain the standard L²-residual formulation. 我們真的需要 H⁻¹-範數 LSFEM 嗎？ 修正通量、保留 L² 殘差，並正確處理混合邊界條件。

  Non-intrusive least-squares functional error estimation Solve with Your Preferred Solver. Estimate with Least Squares. The a posteriori estimator does not depend on your PDE solver. Keep its solution, recover a flux, and use the least-squares functional to estimate the error. 用你喜歡的求解器求解；用最小二乘泛函做後驗誤差估計。 原方程用什麼方法求解都可以；保留近似解、恢復 flux，再由 LS functional 獨立進行後驗誤差估計。
• Singular-source problems: Dirac mixed FEM 奇異源問題 A Dirac Source Is First a Failure of Duality Exact divergence splitting yields a modified-flux formulation; the analysis moves from L² to L^p. Dirac 源首先是對偶失效
• Kellogg interface problem Kellogg 界面問題 Why Is the Kellogg Mesh Radial? A hidden coefficient-weighted energy symmetry explains the ideal adaptive mesh density. 為什麼 Kellogg 網格是徑向的？
• Gibbs phenomena Gibbs 現象 Did the Method Really Resolve the Gibbs Phenomenon? Stable approximation and absence of overshoot are two different questions. 這個方法真的解決了 Gibbs 現象嗎？
• Reduced basis methods 降階基方法
• Neural de Rham approximation 神經 de Rham 逼近

Selected Papers 部分論文

1. Least-Squares Finite Element Methods with Weakly Imposed Boundary Conditions: Norm and Error Equivalences, Over-Penalization, and Fully Discontinuous DiscretizationsShun Zhang · arXiv:2610.17678 [math.NA] · 2026
2. FOSLS-deRhaNN: native de Rham neural classes for H(div) and H(curl) with applications to first-order system least-squares neural network methods for partial differential equationsShun Zhang · arXiv:2610.08016 [math.NA] · 2026
3. Why the Kellogg Mesh Is Radial: A Mathematical Explanation of a Classical Computational BenchmarkShun Zhang · arXiv:2608.22991 · 2026
4. Mixed Finite Element Methods for a Dirac Source: Divergence-Form Splitting and L^p Error AnalysisYueyao Wu and Shun Zhang · arXiv:2608.17575 · 2026
5. Goal-Oriented Error Estimation for Least-Squares Finite Element Methods via Physically Meaningful Adjoint PDEsYueyao Wu and Shun Zhang · arXiv:2607.23850 · 2026
6. Non-intrusive least-squares functional a posteriori error estimator: Linear and nonlinear problems with plain convergenceZiyan Li and Shun Zhang · Computers & Mathematics with Applications 191 (2025), 275–295 · Research page
7. Least-Squares Methods with Nonconforming Finite Elements for General Second-Order Elliptic EquationsYuxiang Liang and Shun Zhang · Journal of Scientific Computing 96 (2023), Article 15
8. A simple proof of coerciveness of first-order system least-squares methods for general second-order elliptic PDEsShun Zhang · Computers & Mathematics with Applications 130 (2023), 98–104
9. Battling Gibbs phenomenon: On finite element approximations of discontinuous solutions of PDEsShun Zhang · Computers & Mathematics with Applications 122 (2022), 35–47

For a fuller publication list, see Google Scholar.