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<p>We present a mixed weak Galerkin finite element method (WG-FEM) for singularly perturbed biharmonic problems on the unit square domain with clamped boundary conditions. By introducing an auxiliary variable, the fourth-order equation is reformulated as a coupled second-order system. A layer-adapted Shishkin mesh is employed to efficiently capture boundary layers. We derive parameter-uniform error bounds in both the energy and balanced norms, proving convergence independent of the perturbation parameter. Numerical results are provided to confirm the theoretical analysis and to demonstrate the effectiveness of the mixed WG-FEM with layer-adapted meshes.</p>

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A weak Galerkin mixed finite element method for singularly perturbed biharmonic problems on a layer-adapted mesh in 2D

  • 期刊论文

  • 发表日期：

    2026年6月16日

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  • Aayushman Raina,
  • Srinivasan Natesan,
  • Şuayip Toprakseven

摘要

We present a mixed weak Galerkin finite element method (WG-FEM) for singularly perturbed biharmonic problems on the unit square domain with clamped boundary conditions. By introducing an auxiliary variable, the fourth-order equation is reformulated as a coupled second-order system. A layer-adapted Shishkin mesh is employed to efficiently capture boundary layers. We derive parameter-uniform error bounds in both the energy and balanced norms, proving convergence independent of the perturbation parameter. Numerical results are provided to confirm the theoretical analysis and to demonstrate the effectiveness of the mixed WG-FEM with layer-adapted meshes.