A hybrid parallel framework to solve convection-diffusion equation using finite element method with variational multiscale stabilization

Standard Galerkin method results in numerical instabilities when applied to the convection-dominated convection–diffusion equations. One approach to address this issue is the variational multiscale (VMS) stabilization technique. However, in the VMS in practice, geometry transformations of the corresponding operators are required, and the diffusion term in the stabilization part involves Christoffel symbols, which do not appear in the classical weak form of the $2^{nd}$-order differential equations. Furthermore, the residual-driven stabilized finite element equation in the VMS method requires integration over multiple terms with different orders of polynomials. Therefore, intensive computational resources are needed to evaluate these terms, which makes the application of this method computationally expensive, especially when high-order elements are used. Optimum parallelization is therefore desirable. This work demonstrates the implementation and verification of the Galerkin approach stabilized using the VMS technique on a hybrid parallel framework with simultaneous use of different parallelization paradigms including shared memory (OpenMP), distributed memory (MPI), and GPGPUs. Load balancing on one heterogeneous computing platform is achieved by offloading the calculations to multiple GPUs, using shared memory parallelism for loops, and distributed memory for linear solvers. Verification of this implementation includes the convergence rate analysis using problems with manufactured solutions, and a benchmark case is solved to compare the convergence rate with other published work. The speed-up data are reported.