R. Altmann

R. Altmann, J. Ramme, P. Schulze

It is well-known that the midpoint rule preserves the dissipation inequality if applied to a certain class of energy-based models. We introduce an appropriate scaling of the state variables such that the symmetric part of the resulting iteration matrix is guaranteed to be positive definite. This allows the application of three-term iteration schemes such as the methods of Widlund and Rapoport. Special emphasis is put on examples where the symmetric part is block diagonal such that the computations decouple. This then leads to efficient dissipation-preserving numerical schemes as illustrated in two numerical examples, namely the biharmonic heat equation and linear poroelasticity.

R. Altmann, A. Moradi

This paper proposes and analyzes an implicit-explicit BDF-Galerkin scheme of second order for the time-dependent nonlinear thermistor problem. For this, we combine the second-order backward differentiation formula with special extrapolation terms for time discretization with standard finite elements for spatial discretization. Unconditionally superclose and superconvergent error estimates are established, relying on two key techniques. First, a time-discrete system is introduced to decompose the error function into its temporal and spatial components. Second, superclose error estimates between the numerical solution and the interpolation of the time-discrete solution are employed to effectively handle the nonlinear coupling term. Finally, we present numerical examples that validate the theoretical findings, demonstrating the unconditional stability and the second-order accuracy of the proposed method.

R. Altmann, R. Maier, J. Schmeck

Within this paper, we introduce partially and fully decoupled time stepping schemes for linear thermo-poroelasticity. This means that the mechanics, heat, and flow equations can be solved sequentially. We provide sufficient conditions on the material parameters, which can be checked a priori, guaranteeing first-order convergence of the introduced schemes. Hence, the proposed methods have the same order as the implicit Euler scheme but are computationally more efficient due to the decoupling of the system equations. Numerical examples validate the proven convergence results and analyze the sharpness of the mentioned parameter condition. Further, we compare the schemes with other decoupling schemes from the literature.