Doubly nonlinear evolution equations of second order: Existence and fully discrete approximation

Abstract

Existence of solutions for a class of doubly nonlinear evolution equations of second order is proven by studying a full discretization. The discretization combines a time stepping on a nonuniform time grid, which generalizes the well-known Störmer-Verlet scheme, with an internal approximation scheme. The linear operator acting on the zero-order term is supposed to induce an inner product, whereas the nonlinear time-dependent operator acting on the first-order time derivative is assumed to be hemicontinuous, monotone and coercive (up to some additive shift), and to fulfill a certain growth condition. The analysis also extends to the case of additional nonlinear perturbations of both the operators, provided the perturbations satisfy a certain growth and a local Hölder-type continuity condition. A priori estimates are then derived in abstract fractional Sobolev spaces. Convergence in a weak sense is shown for piecewise polynomial prolongations in time of the fully discrete solutions under suitable requirements on the sequence of time grids.

On a Full Discretisation for Nonlinear Second-Order Evolution Equations with Monotone Damping: Construction, Convergence, and Error Estimates

“…The following result deduced in LIONS, STRAUSS [26] ensures existence and uniqueness of a solution to the initial value problem (1.1), see also EMMRICH, THALHAMMER [20] for a generalisation obtained via a full discretisation.…”

“…In the present situation, contrary to the convergence analysis given in EMMRICH, THALHAMMER [20] for a full discretisation method based on the one-step backward differentiation formula, we also need to prove strong convergence of the first iterates towards the initial approximations. This is related to EMMRICH [14], where the two-step backward differentiation formula has been studied for the time discretisation of non-Newtonian fluid flows.…”

“…As ensured by Lemma 5.3, it is also possible to establish a strong convergence result for the first iterates. In addition, in order to verify that the obtained limiting function is indeed a solution to the nonlinear second-order evolution equation, we utilise an appropriate analogue to the integration-by-parts formula, which is found in EMMRICH, THALHAMMER [20] and similarly in LIONS, STRAUSS [26]. For the convenience of the reader, we restate this result in Lemma 5.4.…”

“…A convergence result provided in COLLI, FAVINI [9] for a semi-discretisation in time applies to the considerably less involved case, where the domains V A and V B coincide, the nonlinear operator A is maximal monotone, and the linear operator B is bounded, symmetric, as well as strongly positive, up to an additive shift. More recently, convergence of a time discretisation and hence also existence of a weak solution to (1.1) has been proven in EMMRICH, THALHAM-MER [19] under the requirement that V A is dense and continuously embedded in V B , and in EMMRICH, THALHAMMER [20] the convergence analysis has been extended to a full discretisation method, which allows to include problems where V A = V B as well as second-order evolution equations involving non-monotone perturbations and thereby generalises the existence result given in LIONS, STRAUSS [26]. These results have been complemented and illustrated in EMMRICH,ŠIŠKA [16].…”

“…Although the a priori estimates obtained for the fully discrete solution would allow to study functions f ∈ (L p (0, T ;V A )) * + L 1 (0, T ; H * ), we do not exploit this generalisation, see LIONS, STRAUSS [26]. Contrary, the incorporation of nonmonotone perturbations occuring in the description of lower order spatial derivatives seems to be a demanding objective and shall be carried out in a future work, see also EMMRICH, THALHAMMER [20,Section 3].…”