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

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.