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A POD-enhanced simplified weak Galerkin method for solute transport problems

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  • Hooshyarfarzin, Baharak
  • Abbaszadeh, Mostafa
  • Dehghan, Mehdi

Abstract

A simplified weak Galerkin (SWG) method is proposed to solve the solute transport problem. The SWG method makes the possibility of using the arbitrarily shaped polygonal meshes which leads to significant advantages such as fewer elemental calculations compared with the standard finite element method (FEM). This method has important advantages over polygonal finite element method (PFEM), virtual element method (VEM) and usual WGM including no requirement for explicit form of the basis functions, enabling better treatment of problems with discontinuous solutions because of the use of totally discontinuous basis functions and simplicity in constructing discrete schemes. Time discretization is performed using the Crank–Nicolson finite difference method. Convergence error analysis is established in the H1 and L2 norms, showing optimal spatial convergence rates of order O(h) and O(h2), respectively. Furthermore, we combine the SWG method with the proper orthogonal decomposition (POD) technique (POD-SWG) to reduce the dimension of full scheme and the computational complexity of the method. Additionally, a convergence analysis of the POD–SWG plan is established. So, POD-SWG method proposed a computational efficient approach and also had sufficiently high accuracy. Numerical results are given that verify the theoretical analysis and show the computational efficiency and acceptable accuracy of the proposed method.

Suggested Citation

  • Hooshyarfarzin, Baharak & Abbaszadeh, Mostafa & Dehghan, Mehdi, 2026. "A POD-enhanced simplified weak Galerkin method for solute transport problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 247(C), pages 318-346.
    • Handle: RePEc:eee:matcom:v:247:y:2026:i:c:p:318-346
    DOI: 10.1016/j.matcom.2026.03.003
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    References listed on IDEAS

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    1. Zhang, Tie & Chen, Yanli, 2019. "An analysis of the weak Galerkin finite element method for convection–diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 346(C), pages 612-621.
    2. Tao, Xia & Hao, Jiaxiong & Zhang, Yu, 2024. "The uniform convergence of a weak Galerkin finite element method in the balanced norm for reaction–diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 220(C), pages 445-461.
    3. Mazzia, Annamaria, 2020. "A numerical study of the virtual element method in anisotropic diffusion problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 177(C), pages 63-85.
    4. Wang, Shuo & Ma, Jie & Du, Ning, 2025. "A stabilizer-free weak Galerkin finite element method for an optimal control problem of a time fractional diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 231(C), pages 99-118.
    5. Adak, D. & Natarajan, S., 2020. "Virtual element method for semilinear sine–Gordon equation over polygonal mesh using product approximation technique," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 172(C), pages 224-243.
    6. Luo, Zhendong & Li, Hong & Sun, Ping & An, Jing & Navon, Ionel Michael, 2013. "A reduced-order finite volume element formulation based on POD method and numerical simulation for two-dimensional solute transport problems," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 89(C), pages 50-68.
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