We prove that the superconvergence properties of the hybridizable discontinuous Galerkin method for second-order elliptic problems do hold uniformly in time for the semidiscretization by the same method of the heat equation provided the solution is smooth enough. Thus, if the approximations are piecewise polynomials of degree k, the approximation to the gradient converges with the rate h k+1 for k ≥ 0 and the L 2-projection of the error into a space of lower polynomial degree superconverges with the rate log(T /h 2) h k+2 for k ≥ 1 uniformly in time. As a consequence, an elementby-element postprocessing converges with the rate log(T /h 2) h k+2 for k ≥ 1 also uniformly in time. Similar results are proven for the Raviart-Thomas and the Brezzi-Douglas-Marini mixed methods.

We consider a gel as an immiscible mixture of polymer and solvent, and derive governing equations of the dynamics. They include the balance of mass and linear momentum of the individual components. The model allows to account for nonlinear elasticity, viscoelasticity, transport and diffusion. The total free energy of the system combines the elastic contribution of the polymer with the FloryHuggins energy of mixing. The system is also formulated in terms of the center of mass velocity and the diffusive velocity, involving the total and the relative stresses. This allows for the identification of special regimes, such as the purely diffusive and the transport ones. We also obtain an equation for the rate of change of the total energy yielding decay for special choices of boundary conditions. The energy law motivates the Rayleghian variational approach discussed in the last part of the article. We consider the case of a gel in a one-dimensional strip domain in order to study special features of the dynamics, in particular, the early dynamics. We find that the monotonicity of the extensional stress is a necessary condition to guarantee the propagation of the swelling interface between the gel and its solvent. Such monotonicity condition is satisfied for data corresponding to linear entangled polymers. However, for polyssacharide gels the monotonicity of the stress fails at a critical volume fraction, suggesting the onset of de-swelling. The weak elasticity is responsible for the loss of monotonicity of the stress. The analysis also suggests that type II diffusion is a hyperbolic phenomenon rather than a diffusive one. One goal is to compare the derivation method, assumptions and resulting equations with other models available in the literature, and determine their regimes of validity. The stress-diffusion coupling model by Yamaue and Doi is one main benchmark. We assume that the gel is non-ionic, and neglect thermal effects.

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We develop a hybrid implicit and explicit adaptive multirate time integration method to solve systems of time-dependent equations that present two significantly different scales. We adopt an iteration scheme to decouple the equations with different time scales. At each iteration, we use an implicit Galerkin method with a fast time-step to solve for the fast scale variables and an explicit method with a slow time-step to solve for the slow variables. We derive an error estimator using a posteriori analysis which controls both the iteration number and the adaptive time-step selection. We present several numerical examples demonstrating the efficiency of our scheme and conclude with a stability analysis for a model problem.Keywords Multirate equations · Explicit and implicit methods · A posteriori error analysis · Stability