Nonlinear discontinuous Petrov–Galerkin methods

Abstract

The discontinuous Petrov-Galerkin method is a minimal residual method with broken test spaces and is introduced for a nonlinear model problem in this paper. Its lowest-order version applies to a nonlinear uniformly convex model example and is equivalently characterized as a mixed formulation, a reduced formulation, and a weighted nonlinear least-squares method. Quasi-optimal a priori and reliable and efficient a posteriori estimates are obtained for the abstract nonlinear dPG framework for the approximation of a regular solution. The variational model example allows for a built-in guaranteed error control despite inexact solve. The subtle uniqueness of discrete minimizers is monitored in numerical examples. Classification (2000) 47H05,49M15,65N12,65N15,65N30 Reliable simulation techniques in solid mechanics. Development of non-standard discretization methods, mechanical and mathematical analysis' under the project 'Foundation and application of generalized mixed FEM towards nonlinear problems in solid mechanics' (CA Mathematics Subject

A DPG Framework for Strongly Monotone Operators

“…This, and extensions to more general nonlinear problems, is ongoing research. In contrast, we do not see an obvious extension of [5] or [6] to singularly perturbed nonlinear problems, except for introducing a linear relaxation as we propose.…”

“…We analyze approximations of the continuous problems (5) and (6). At the continuous level, these two problems are equivalent by Theorem 2.3 so that considering T κ or D κ with their respective right-hand side yields the same problem.…”

“…The discrete operator D κ,h defines a conforming approximation of the continuous problem (6). Therefore, its uniform Lipschitz continuous invertibility follows from the Lipschitz continuous invertibility of D κ , cf.…”

“…In the context of the nonlinear DPG scheme from [6] we mentioned their extension to a nonlinear mixed form. In fact, in the linear case it is well known that there is a mixed form of the DPG scheme, and this is precisely the method proposed (for a specific model problem) by Cohen et al [11].…”

“…A different idea is to apply the minimum residual technique in product or "broken" spaces to the nonlinear problem. Bui-Thanh and Ghattas [5] did this by considering the entire nonlinear problem as a constraint, and Carstensen et al [6] developed a representation of the DPG scheme by a nonlinear mixed form and analyzed the case of lowest order approximations. DPG for contact problems has been studied in [21], though in this case the nonlinearity is due to the contact condition which is treated by a variational inequality.…”