13 January – Koondanibha Mitra : A posteriori estimates for nonlinear degenerate parabolic and elliptic equations

Koondanibha Mitra Thursday 13th January at 11:00



Nonlinear advection-diffusion-reaction equations are used to model various complex flow processes in porous media, and in biological systems. They also exhibit parabolic-hyperbolic and parabolic-elliptic kinds of degeneracies resulting in the loss of regularity of the solutions. The nonlinear degenerate nature of the equations makes it challenging to provide sharp error bounds to any numerical solutions of the problem. When discretized in time, such equations result in a sequence of nonlinear degenerate elliptic problems which requires linear iterative schemes to solve. The linear iterates can be used to provide upper/lower bounds to the error, and to separate the error contributions due to linearization and discretization. However, the nonlinearity, as before, impedes the derivation of sharp error bounds in the standard error norm.

In the first part of this study, we provide reliable, fully computable, and locally space-time efficient a posteriori error bounds for numerical approximations of such nonlinear degenerate parabolic problems. For showing global reliability, a nonlocal-in-time error estimate is derived individually for the time-integrated $H^1(H^{-1})$, $L^2(L^2)$, and the $L^2(H^1)$ 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 $H^1(H^{-1})\cap L^2(H^1)$ norm. The reliability and efficiency norms employed coincide when there is no nonlinearity. Moreover, error contributors such as flux nonconformity, time discretization, quadrature, and data oscillation are identified and separated. Numerical tests are conducted for nondegenerate and degenerate cases having exact solutions, as well as for realistic cases. It is shown that the estimators correctly identify the errors up to a factor of the order of unity.

In the second part, using linear iterative schemes, we derive reliable, fully computable, and efficient error bounds for the finite element solution of the elliptic problem which originates from the time-discretization of the parabolic equation. For obtaining sharp bounds, a pseudo-norm is introduced which is invoked by the linear operator associated with the iterative scheme. The equivalence between a standard norm and the pseudo-norms are shown. An orthogonality relation is derived equating the error with a linearization component and a discretization component. This equality relation is then used to bind from above and below the error, using computable a posteriori estimators.

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