
arXiv: 2108.12507
The Richards equation is commonly used to model the flow of water and air through soil, and it serves as a gateway equation for multiphase flows through porous media. It is a nonlinear advection–reaction–diffusion equation that exhibits both parabolic–hyperbolic and parabolic–elliptic kind of degeneracies. In this study, we provide reliable, fully computable, and locally space–time efficient a posteriori error bounds for numerical approximations of the fully degenerate Richards equation. For showing global reliability, a nonlocal-in-time error estimate is derived individually for the time-integrated H 1 ( H − 1 ) H^1(H^{-1}) , L 2 ( L 2 ) L^2(L^2) , and the L 2 ( H 1 ) L^2(H^1) errors. A maximum principle and a degeneracy estimator are employed for the last one. Global and local space–time efficiency error bounds are then obtained in a standard H 1 ( H − 1 ) ∩ L 2 ( H 1 ) H^1(H^{-1})\cap L^2(H^1) norm. The reliability and efficiency norms employed coincide when there is no nonlinearity. Moreover, error contributors such as space discretization, time discretization, quadrature, linearization, and data oscillation are identified and separated. The estimates are also valid in a setting where iterative linearization with inexact solvers is considered. Numerical tests are conducted for nondegenerate and degenerate cases having exact solutions, as well as for a realistic case and a benchmark case. It is shown that the estimators correctly identify the errors up to a factor of the order of unity.
a posteriori error estimates, A posteriori error estimates, Finite element method, Flows in porous media; filtration; seepage, Flow through porous media, finite element method, Richards equation, Numerical Analysis (math.NA), [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs, PDEs in connection with fluid mechanics, Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs, Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.), flow through porous media, Error bounds for initial value and initial-boundary value problems involving PDEs, nonlinear degenerate problems, Finite difference methods for initial value and initial-boundary value problems involving PDEs, FOS: Mathematics, Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs, Mathematics - Numerical Analysis, Nonlinear degenerate problems, 65M15, 65N50
a posteriori error estimates, A posteriori error estimates, Finite element method, Flows in porous media; filtration; seepage, Flow through porous media, finite element method, Richards equation, Numerical Analysis (math.NA), [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs, PDEs in connection with fluid mechanics, Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs, Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.), flow through porous media, Error bounds for initial value and initial-boundary value problems involving PDEs, nonlinear degenerate problems, Finite difference methods for initial value and initial-boundary value problems involving PDEs, FOS: Mathematics, Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs, Mathematics - Numerical Analysis, Nonlinear degenerate problems, 65M15, 65N50
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