Quasi-Newton variable preconditioning for nonlinear nonuniformly monotone elliptic problems posed in Banach spaces

Quasi-Newton-type iterative solvers are developed for a wide class of nonlinear elliptic problems. The presented generalization of earlier efficient methods covers various nonuniformly elliptic problems arising, e.g., in non-Newtonian flows or for certain glaciology models. The robust estimates are reinforced by several examples

Sobolev gradient preconditioning for the electrostatic potential equation

AbstractSobolev gradient type preconditioning is proposed for the numerical solution of the electrostatic potential equation. A constructive representation of the gradients leads to efficient Laplacian preconditioners in the iteration thanks to the available fast Poisson solvers. Convergence is then verified for the corresponding sequence in Sobolev space, implying mesh independent convergence results for the discretized problems. A particular study is devoted to the case of a ball: due to the radial symmetry of this domain, a direct realization without discretization is feasible. The simplicity of the algorithm and the fast linear convergence are finally illustrated in a numerical test example

Discrete maximum principles for nonlinear parabolic PDE systems

Discrete maximum principles (DMPs) are established for finite element approximations of systems of nonlinear parabolic partial differential equations with mixed boundary and interface conditions. The results are based on an algebraic DMP for suitable systems of ordinary differential equations

Variable preconditioning for strongly nonlinear elliptic problems

Variable preconditioning has earlier been developed as a realization of quasi-Newton methods for elliptic problems with uniformly bounded nonlinearities. This paper presents a generalization of this approach to strongly nonlinear problems, first on an operator level, then for elliptic problems allowing power order growth of nonlinearities. Numerical tests reinforce the convergence results

Newton’s Method in the Context of Gradients

This article gives a common theoretical treatment for gradient and Newton type methods for general classes of problems

The role of water and compression in the genesis of alkaline basalts: Inferences from the Carpathian-Pannonian region

We present a new model for the formation of Plio-Pleistocene alkaline basalts in the central part of the Carpathian-Pannonian region (CPR). Based on the structural hydroxyl content of clinopyroxene megacrysts, the ‘water’ content of their host basalts is 2.0–2.5 wt.%, typical for island arc basalts. Likewise, the source region of the host basalts is ‘water’ rich (290–660 ppm), akin to the source of ocean island basalts. This high ‘water’ content could be the result of several subduction events from the Mesozoic onwards (e.g. Penninic, Vardar and Magura oceans), which have transported significant amounts of water back to the upper mantle, or hydrous plumes originating from the subduction graveyard beneath the Pannonian Basin. The asthenosphere with such a relatively high ‘water’ content beneath the CPR may have been above the ‘pargasite dehydration’ (90 km) solidi. This means that neither decompressional melting nor the presence of voluminous pyroxenite and eclogite lithologies are required to explain partial melting. While basaltic partial melts have been present in the asthenosphere for a long time, they were not extracted during the syn-rift phase, but were only emplaced at the onset of the subsequent tectonic inversion stage at ~8–5 Ma. We propose that the extraction has been facilitated by evolving vertical foliation in the asthenosphere as a response to the compression between the Adriatic indenter and the stable European platform. The vertical foliation and the prevailing compression effectively squeezed the partial basaltic melts from the asthenosphere. The overlying lithosphere may have been affected by buckling in response to compression, which was probably accompanied by formation of deep faults and deformation zones. These zones formed conduits towards the surface for melts squeezed out of the asthenosphere. This implies that basaltic partial melts could be present in the asthenosphere in cases where the bulk ‘water’ content is relatively high (>~200 ppm) at temperatures exceeding ~1000–1100 °C. These melts could be extracted even under a compressional tectonic regime, where the combination of vertical foliation in the asthenosphere and deep fractures and deformation zones in the folded lithosphere provides pathways towards the surface. This model is also valid for deep seated transpressional or transtensional fault zones in the lithosphere