InterPore2021

A posteriori estimates for the Richards equation (nonlinearity, degeneracy & heterogeneity)

2 Jun 2021, 19:50

15m

Oral Presentation (MS7) Mathematical and numerical methods for multi-scale multi-physics, nonlinear coupled processes MS7

Speaker

Dr Koondanibha Mitra (Radboud University Nijmegen)

Description

Richards equation is commonly used to model the flow of water and air through soil, and it serves as a gateway equation for multiphase flow through porous domains. With pressure $p$ being the primary variable, it equates
$$\begin{array}{}\text{(1)}& {\partial }_{t}S(p)-\mathrm{\nabla }\cdot [\overline{\mathbf{K}}\kappa (S(p))(\mathrm{\nabla }p-\mathbf{g})]=f(s,\mathbf{x},t),\end{array}$$
where $S:[-\mathrm{\infty },\mathrm{\infty }]\to [0,1]$ is the increasing saturation function, $\kappa :[0,1]\to [0,1]$ is the relative permeability function, $\overline{\mathbf{K}}$ the absolute permeability tensor, $\mathbf{g}$ the gravitational acceleration, and $f$ the reaction/absorption term. Apart from having nonlinear advection/reaction/diffusion components, Richards equation also exhibits both parabolic-hyperbolic (at $S(p)=0$ since $\kappa (S(p))=0$) and parabolic-elliptic (at $S(p)=1$ since $S^{\prime }(p)=0$) type of degeneracies. Further challenges in its numerical treatment comes from the heterogeneity in $\overline{\mathbf{K}}$.

In this study, we provide fully computable, locally space-time efficient, and reliable a posteriori error bounds [1] for numerical solutions of the fully degenerate Richards equation: if $p$ is the exact solution of (1) and $p_{h\tau }$ is a known approximate solution, then for a composite distance metric $\mathrm{dist}(\cdot ,\cdot )$ it holds that
$$\begin{array}{}\text{(2)}& \underset{―}{\eta }(p_{h\tau })\le \mathrm{dist}(p,p_{h\tau })\le \overline{\eta }(p_{h\tau }),\end{array}$$
where both $\underset{―}{\eta }(\cdot )$ and $\overline{\eta }(\cdot )$ are fully computable, and a version of the lower bound holds in any space-time subdomain. The bounds are proven in a variation of the $H^1(H^{-1})\cap L^2(L^2)\cap L^2(H^1)$ norm which corresponds to the minimum regularity inherited by the exact solutions, thus avoiding further smoothness assumptions like in [2]. For showing the upper bound, error estimates are derived individually for the $H^1(H^{-1})$, $L^2(L^2)$ and the $L^2(H^1)$ error components with a maximum principle and a novel degeneracy estimator being used for the last one. Local and global space-time efficient error bounds are obtained following [3]. Error contributors such as flux and time non-conformity, quadrature, linearisation, data oscillation are identified and separated. The estimates also work in a fully adaptive space-time discretization and linearisation setting.

To investigate the effectiveness of the estimators, numerical tests are conducted for non-degenerate and degenerate cases having exact solutions. It is shown that the estimators correctly identify the errors, both spatially and temporally, up to a factor in the order of unity. Finally, to demonstrate the prowess of the estimators, a degenerate problem is analyzed in a heterogeneous, anisotropic domain with discontinuous initial condition and mixed boundary conditions.

References

[1] M. Ainsworth and J.T. Oden. A posteriori error estimation in ?finite element analysis. Pure and Applied Mathematics (New York). Wiley-Interscience [John Wiley & Sons],
New York, 2000

[2] V.Dolejší, A. Ern, and M. Vohralík. A framework for robust a posteriori error control in unsteady nonlinear advection-diffusion problems. SIAM Journal on Numerical Analysis, 51(2): 773-793, 2013.

[3] A. Ern, I. Smears, and M. Vohralík. Guaranteed, locally space-time efficient, and polynomial-degree robust a posteriori error estimates for high-order discretizations of parabolic problems. SIAM Journal of Numerical Analysis, 55(6): 2811-2834, 2017.

Presentation materials

Figure_Abstract_KMITRA.jpg