A symmetric nodal conservative finite element method for the Darcy equation

This work introduces and analyzes novel stable Petrov-Galerkin EnrichedMethods (PGEM) for the Darcy problem based on the simplest but unstable continuous P1/P0 pair. Stability is recovered inside a Petrov-Galerkin framework where element-wise dependent residual functions, named multi-scale functions, enrich both velocity and pressure trial spaces. Unlike the velocity test space that is augmented with bubble-like functions, multi-scale functions correct edge residuals as well. The multi-scale functions turn out to be the well-known lowest order Raviart-Thomas basis functions for the velocity and discontinuous quadratics polynomial functions for the pressure. The enrichment strategy suggests the way to recover the local mass conservation property for nodal-based interpolation spaces. We prove that the method and its symmetric version are well-posed and achieve optimal error estimates in natural norms. Numerical validations confirm claimed theoretical results

Postprocessing of Non-Conservative Flux for Compatibility with Transport in Heterogeneous Media

A conservative flux postprocessing algorithm is presented for both steady-state and dynamic flow models. The postprocessed flux is shown to have the same convergence order as the original flux. An arbitrary flux approximation is projected into a conservative subspace by adding a piecewise constant correction that is minimized in a weighted $L^2$ norm. The application of a weighted norm appears to yield better results for heterogeneous media than the standard $L^2$ norm which has been considered in earlier works. We also study the effect of different flux calculations on the domain boundary. In particular we consider the continuous Galerkin finite element method for solving Darcy flow and couple it with a discontinuous Galerkin finite element method for an advective transport problem.Comment: 34 pages, 17 figures, 11 table

Consistent local projection stabilized finite element methods

This work establishes a formal derivation of local projection stabilized methods as a result of an enriched Petrov-Galerkin strategy for the Stokes problem. Both velocity and pressure finite element spaces are enhanced with solutions of residual-based local problems, and then the static condensation procedure is applied to derive new methods. The approach keeps degrees of freedom unchanged while gives rise to new stable and consistent methods for continuous and discontinuous approximation spaces for the pressure. The resulting methods do not need the use of a macro-element grid structure and are parameter-free. The numerical analysis is carried out showing optimal convergence in natural norms, and moreover, two ways of rendering the velocity field locally mass conservative are proposed. Some numerics validate the theoretical results

Stabilization arising from PGEM : a review and further developments

The aim of this paper is twofold. First, we review the recent Petrov-Galerkin enriched method (PGEM) to stabilize numerical solutions of BVP's in primal and mixed forms. Then, we extend such enrichment technique to a mixed singularly perturbed problem, namely, the generalized Stokes problem, and focus on a stabilized finite element method arising in a natural way after performing static condensation. The resulting stabilized method is shown to lead to optimal convergences, and afterward, it is numerically validated

Enriched Galerkin method for the shallow-water equations

This work presents an enriched Galerkin (EG) discretization for the two-dimensional shallow-water equations. The EG finite element spaces are obtained by extending the approximation spaces of the classical finite elements by discontinuous functions supported on elements. The simplest EG space is constructed by enriching the piecewise linear continuous Galerkin space with discontinuous, element-wise constant functions. Similarly to discontinuous Galerkin (DG) discretizations, the EG scheme is locally conservative, while, in multiple space dimensions, the EG space is significantly smaller than that of the DG method. This implies a lower number of degrees of freedom compared to the DG method. The EG discretization presented for the shallow-water equations is well-balanced, in the sense that it preserves lake-at-rest configurations. We evaluate the method’s robustness and accuracy using various analytical and realistic benchmarks and compare the results to those obtained using the DG method. Finally, we briefly discuss implementation aspects of the EG method within our MATLAB / GNU Octave framework FESTUNG

A subcell-enriched Galerkin method for advection problems

In this work, we introduce a generalization of the enriched Galerkin (EG) method. The key feature of our scheme is an adaptive two-mesh approach that, in addition to the standard enrichment of a conforming finite element discretization via discontinuous degrees of freedom, allows to subdivide selected (e.g. troubled) mesh cells in a non-conforming fashion and to use further discontinuous enrichment on this finer submesh. We prove stability and sharp a priori error estimates for a linear advection equation by using a specially tailored projection and conducting some parts of a standard convergence analysis for both meshes. By allowing an arbitrary degree of enrichment on both, the coarse and the fine mesh (also including the case of no enrichment), our analysis technique is very general in the sense that our results cover the range from the standard continuous finite element method to the standard discontinuous Galerkin (DG) method with (or without) local subcell enrichment. Numerical experiments confirm our analytical results and indicate good robustness of the proposed method