Abstract
Variational time discretization schemes are getting of increasing importance for the accurate numerical approximation of transient phenomena. The applicability and value of mixed finite element methods in space for simulating transport processes have been demonstrated in a wide class of works. We consider a family of continuous Galerkin–Petrov time discretization schemes that is combined with a mixed finite element approximation of the spatial variables. The existence and uniqueness of the semidiscrete approximation and of the fully discrete solution are established. For this, the Banach–Nečas–Babuška theorem is applied in a non-standard way. Error estimates with explicit rates of convergence are proved for the scalar and vector-valued variable. An optimal order estimate in space and time is proved by duality techniques for the scalar variable. The convergence rates are analyzed and illustrated by numerical experiments, also on stochastically perturbed meshes.
Highlights
Numerical simulations of time dependent single and multiphase phase flow and multicomponent transport processes in complex and porous media with strong heterogeneities and anisotropies are desirable in several fields of natural sciences and civil engineering as well as in a large number of branches of technology; cf. e.g., [22,29]
Variational time discretization schemes that are combined with continuous or discontinuous finite element methods for the spatial variables are studied for flow and parabolic problems in, for instance [1,2,3,4,10,15,30,31,32,38,47] and for wave problems in, for instance [7,36,37]
As a prototype model for more sophisticated multiphase flow and multicomponent reactive transport systems in porous media we study in this work
Summary
Numerical simulations of time dependent single and multiphase phase flow and multicomponent transport processes in complex and porous media with strong heterogeneities and anisotropies are desirable in several fields of natural sciences and civil engineering as well as in a large number of branches of technology; cf. e.g., [22,29]. Variational time discretization schemes that are combined with continuous or discontinuous finite element methods for the spatial variables are studied for flow and parabolic problems in, for instance [1,2,3,4,10,15,30,31,32,38,47] and for wave problems in, for instance [7,36,37].
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