A domain-specific language for the hybridization and static condensation of finite element methods

In this paper, we introduce a domain-specific language (DSL) for concisely expressing localized linear algebra on finite element tensors and its integration within a code-generation framework. This DSL is general enough to facilitate the automatic generation of element-based dense linear algebra kernels necessary for the implementation of static condensation methods and local solvers for a variety of problems. We demonstrate its use for the static condensation of continuous Galerkin problems, and systems arising from hybridizing finite element discretizations. We also describe how this DSL can be used to execute local post-processing procedures to construct superconvergent approximations to mixed problems. This work also features highlevel implementations of static condensation and hybrid-mixed methods as preconditioners conforming to PETSc’s interface for solving linear systems. These preconditioning interfaces provide reduced operators, which are obtained from locally assembled expressions, with the necessary context to specify full solver configurations on the resulting linear systems. We validate our implementations within the context of model second-order elliptic problems. Finally, we conclude with a performance comparison highlighting the use of a hybrid-mixed method as an efficient preconditioner in a semi-implicit method for the nonlinear rotating shallow water equations.