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Numerical resolution of advection-diffusion equations on connected graphs requires careful treatment in implementing internal boundary conditions at junctions. This study proposes a simple conforming Petrov-Galerkin finite element method with compact stencil, referred to as CPGFEM2, which effectively solves the non-conservative advection-diffusion equations on connected graphs. The CPGFEM2 utilizes the fitting technique in the spatial discretization so that the junctions are consistently handled as implicit internal boundary conditions. A selective lumping algorithm in conjunction with the local  -scheme is applied in the temporal discretization. The CPGFEM2 is verified through several test problems. The CPGFEM2 is successfully applied to numerical analysis of conservative solute transport in an existing agricultural drainage system, showing its applicability to real problems.

A Conforming Finite Element Method for Non-conservative Advection-diffusion Equations on Connected Graphs