We analyze the existence of solutions for the finite-element discretized stationary and time-dependent sedimentation-consolidation partial differential equations (PDEs) using recent advances in the local-projection stabilization (LPS) method. The sedimentation model studied here couples gravitationally forced Stokes flow and a convection-diffusion equation for the solids concentration, with boundary conditions selected to mimic physical applications. This system is highly nonlinear and sensitive due to the non-constant velocity and concentration coupling terms, as well as the nonlinear flux term in the convection equation. We provide an overview of pre-existing methods already developed for solving this class of problem, establish the coercivity of the underlying operator, as well as conditions for the existence of discrete solutions. For convection-dominated regimes using non-inf-sup stable finite elements with LPS, we demonstrate that stabilization effectively eliminates interior and boundary layers in the velocity and pressure solutions. Numerical examples in two and three dimensions for the non-stationary case illustrate the theoretical results.

This paper shows that the Stokes problem is well-posed when velocity and pressure simultaneously vanish on the domain boundary. This result is achieved by extending Nečas’ inequality to square-integrable functions that vanish in a small band covering the boundary. It is found that the associated a priori pressure estimate depends inversely on the volume of the band. Numerical experiments using the finite element method confirm these findings. Based on these results, guidelines are provided for applying vanishing pressure boundary conditions in model coupling and domain decomposition methods.

This paper addresses the challenge of proving the existence of solutions for nonlinear equations in Banach spaces, focusing on the Navier-Stokes equations and discretizations of thom. Traditional methods, such as monotonicity-based approaches and fixed-point theorems, often face limitations in handling general nonlinear operators or finite element discretizations. A novel concept, mapped coercivity, provides a unifying framework to analyze nonlinear operators through a continuous mapping. We apply these ideas to saddle-point problems in Banach spaces, emphasizing both infinite-dimensional formulations and finite element discretizations. Our analysis includes stabilization techniques to restore coercivity in finite-dimensional settings, ensuring stability and existence of solutions. For linear problems, we explore the relationship between the inf-sup condition and mapped coercivity, using the Stokes equation as a case study. For nonlinear saddle-point systems, we extend the framework to mapped coercivity via surjective mappings, enabling concise proofs of existence of solutions for various stabilized Navier-Stokes finite element methods. These include Brezzi-Pitkäranta, a simple variant, and local projection stabilization (LPS) techniques, with extensions to convection-dominant flows. The proposed methodology offers a robust tool for analyzing nonlinear PDEs and their discretizations, bypassing traditional decompositions and providing a foundation for future developments in computational fluid dynamics.

Sediment accumulation rate is recognized as the primary parameter influencing the burial rate of organic carbon and other compounds in marine sediments. The prediction of a global map for burial rates is challenging due to the limited availability of measurements for total organic carbon (TOC) and sediment accumulation rates from the seafloor. Recent advancements in machine learning, including techniques such as K nearest Neighbours and Random Forests, have demonstrated promise in producing comprehensive predictions utilizing global maps of oceanic properties. In this study, we introduce a sophisticated approach based on a newly developed deep neural network (DNN) model tailored for geospatial predictions. Employing few-shot learning techniques, such as the incorporation of prior physical knowledge into the model, along with strategies like multi-task learning and semi-supervised learning, enhances predictions amidst sparse data availability. Moreover, p​​​​​redictions of the global distribution of seafloor TOC and sediment accumulation rates presented here are coupled with uncertainty maps computed using Monte Carlo Dropout, a Bayesian approximation method that effectively inform about the degree of the model predictibility. With our results, we not only explore the global distribution of burial rates of organic carbon but also offer insights into the global carbon stocks in various marine regions.

The inf-sup condition is one of the essential tools in the analysis of the Stokes equations and especially in numerical analysis. In its usual form, the condition states that for every pressure p∈ L^2(Ω)∖ℝ, (i.e. with mean value zero) a velocity u∈ H^1_0(Ω)^d can be found, so that (div u,p)=p^2 and ∇ u≤ c p applies, where c>0 does not depend on u and p. However, if we consider domains that have a Neumann-type outflow condition on part of the boundary Γ_N⊂∂Ω, the inf-sup condition cannot be used in this form, since the pressure here comes from L^2(Ω) and does not necessarily have zero mean value. In this note, we derive the inf-sup condition for the case of outflow boundaries.

We provide a concise proof of existence of the solutions to nonlinear operator equations in separable Banach spaces, without assuming the operator to be monotone. Instead, our main hypotheses consist of a continuity assumption and a mapped coercivity property, which is a generalization of the usual coercivity property for nonlinear operators. In the case of linear operators, we recover the traditional infsup condition. To illustrate the applicability of this general concept, we apply it to semi-linear elliptic problems and the Navier-Stokes equations. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

Marine protected areas (MPAs) have become the de-facto mechanism for marine species conservation and management and the ecological benefits of no take zones (NTZs) are supported by an expanding scientific interest in their effectiveness. However, in the south-western Baltic region there is a paucity of studies on the validity of MPAs, possibly due to the lack of enforcement. In this work we investigate the effects of nonconstant harvesting of a resource simultaneously under predation pressure. We use a functional response of the predator dependent Crowley-Martin type in age-structured state equations of a fixed time-horizon optimal control problem, wherein a pre-existing MPA may be encountered. Where available, only peer-reviewed data and biomass stock levels are used for biological parameters in the state equations. Our results for the prey component of the model show that a moderate penalty exists that yields the same benefits as a fully closed reserve. However, taking into account the whole system we observe that an NTZ benefits the predator species too.

Spatial predictions of total organic carbon (TOC) concentrations and stocks are crucial for understanding marine sediments’ role as a significant carbon sink in the global carbon cycle. In this study, we present a geospatial prediction of global TOC concentrations and stocks on a 5 × 5 arcmin grid, using a novel neural network approach. We also provide and apply a new compilation of over 21 000 global TOC measurements and a new set of predictors, including features such as seafloor lithologies, benthic oxygen fluxes, and chlorophyll-a satellite data. Moreover, we compare different machine learning models based on their performance metrics and predictions and assess their strengths and limitations. For the dataset used, we find that the performance metrics of the models are comparable and that the neural network approach outperforms, on unseen data, methods such as k-nearest neighbours and random forests, which tend to overfit the training data. We provide estimates of mean TOC concentrations and stocks, both on continental shelves and in deep-sea settings across various marine regions and oceans. Our model suggests that the upper 10 cm of oceanic sediments harbour approximately 156 Pg of TOC stocks and have a mean TOC concentration of 0.61 %. Furthermore, we introduce a standardized methodology for quantifying predictive uncertainty using Monte Carlo dropout. The method was applied to our neural network model and underlying features to generate a map of information gain that measures the expected increase in model knowledge, achieved through additional sampling at specific locations, which is pivotal for sampling strategy planning.

We create a deep neural network based approach for the geospatial predicition of total organic carbon percentages in marine sediments. The code in the repository includes jupyter notebooks and python files to pre-process the data, train the models and post-process the outputs.

Mass accumulation rates of sediments[g/cm2/yr] or sedimentation rates[cm/yr] on the seafloor are important to understand various benthic properties, like the rate of carbon sequestration in the seafloor and seafloor geomechanical stability. Several machine learning models, such as random forests, and k-Nearest Neighbours have been proposed for the prediction of geospatial data in marine geosciences, but face significant challenges such as the limited amount of labels for training purposes, skewed data distribution, a large number of features etc. Previous model predictions show deviation in the global sediment budget, a parameter used to determine a model's predicitve validity, revealing the lack of accurate representation of sedimentation rate by the state of the art models. Here we present a semi-supervised deep learning methodology to improve the prediction of sedimentation rates, making use of around 9x106 unlabelled data points. The semi-supervised neural network implementation has two parts: an unsupervised pretraining using an encoder-decoder network. The encoder with the optimized weights from the unsupervised training is then taken out and fitted with layers that lead to the target dimension. This network is then fine-tuned with 2782 labelled data points, which are observed sedimentation rates from peer-reviewed sources. The fine-tuned model then predicts the rate and quantity of sediment accumulating on the ocean floor, globally.The developed semi-supervised neural network provide better predictions than supervised models trained only on labelled data. The predictions from the semi-supervised neural network are compared with that of the supervised neural network with and without dimensionality reduction(using Principle Component Analysis).

This paper presents a numerical study of using Rosenbrock-type methods for the temporal discretization of the incompress-ible Navier-Stokes equations. We focus on Rosenbrock-Wanner (ROW) and W-methods. In these methods, the treatment of the Navier-Stokes nonlinearity is built directly into the time-stepping schemes. Hence, in each time step, s merely linear systems have to be solved, where s is the number of stages of these methods.While in ROW methods the system matrix is different in each time step, W-methods grant the possibility of keeping the system matrix fixed for some or even all time steps.In this paper, we conduct numerical studies in which we examine four different Rosenbrock-type methods in their ROW method variants and several different W-method variants. We study a manufactured example with known solution and the well-known benchmark problem "flow around a circular obstacle".(c) 2022 Elsevier B.V. All rights reserved.

This paper considers pure strategy Nash equilibria of non-cooperative legislative bargaining models. In contrast to existing legislative bargaining models, we derive legislators behavior from stochastic utility maximization. This approach allows us to prove the existence of a stationary Pure Local and Global Nash Equilibrium under rather general settings. The mathematical proof is based on a fixed point argument, which can also be used as a numerical method to determine an equilibrium. We characterize the equilibrium outcome as a lottery of legislators’ proposals and prove a Mean Voter Theorem , i.e., proposals result dimension-by-dimension as a weighted mean of legislators’ ideal points and are Pareto-optimal. Based on a simple example, we illustrate different logic of our model compared to mixed strategy equilibrium of the legislative bargaining model suggested by Banks and Duggan (Am Polit Sci Rev 94(1):73–88. https://doi.org/10.2307/2586381 , 2000).

We present a short existence proof for nonlinear operator equations in separable Banach spaces. The operator is not supposed to be monotone. The main hypotheses are a continuity assumption and a generalized coercivity property. For illustration, we apply this general concept to semi-linear elliptic problems and to the Navier-Stokes equations.

This paper presents a novel two-level finite element method for convection–diffusion–reaction problems. The proposed scheme consists of a global problem and an ensemble of local problems. The boundary conditions of the local problems are provided by a global approximation of the same problem. The stability is ensured with an artificial diffusion mechanism that acts on the difference between local and global approximation, and leads to an a priori error estimate for the global solution. For the computation of the solutions, an efficient fixed-point algorithm is proposed as an alternative to the well established static condensation technique. The local solutions can be computed in parallel and enter in a communication step. This step takes place in the entire domain and consists of a mass matrix inversion only. Therefore, the overall algorithm is easy to implement and computationally inexpensive.

Abstract This short article serves as an epilog of the thirteen preceding papers in this special issue of CMAM. All contributions are authored by participants of the 7th Sino–German Workshop on Computational and Applied Mathematics at the Kiel University. The topics cover fourth-order problems, solvers and multilevel methods, a posteriori error control and adaptivity, and data science.

This article presents a novel local pressure correction method for incompressible fluid flows and documents a numerical study of this method. Pressure correction methods decouple the velocity and pressure components of the time‐dependent Navier‐Stokes equations and lead to a sequence of elliptic partial differential equations for both components instead of a saddle point problem. In some situations, the equations for the velocity components are solved explicitly (with time step restrictions) and thus the elliptic pressure problem remains to be the most expensive step. Here, we employ a multiscale procedure for the solution of the Poisson problem related to pressure. The procedure replaces the global Poisson problem by local Poisson problems on subregions. We propose a new Robin‐type boundary condition design for the local Poisson problems, which contains a coarse approximation of the global Poisson problem. Accordingly, no further communication between subregions is necessary and the method is perfectly adapted for parallel computations. Numerical experiments regarding a known analytical solution and flow around cylinder benchmarks show the effectivity of this new local pressure correction method.

Mantle convection and melt migration are important processes for understanding Earth’s dynamics and how they relate to observations at the surface. Recently it has been established that melt migration can be modelled by coupling variable-viscosity Stokes flow and Darcy flow, where Stokes flow generally captures the long-term behaviour of the mantle and lithosphere, and Darcy flow models the two-phase regime. It is known that approximating the solution by finite element methods requires the use of mixed inf-sup stable elements or additional stabilization terms. Here, we propose a formulation with a coercive non-symmetric linear operator which allows the use of simple equal-order elements.

Pressure correction methods constitute the most widely used solvers for the time-dependent Navier-Stokes equations. There are several different pressure correction methods, where each time step usually consists in a predictor step for a non-divergence-free velocity, followed by a Poisson problem for the pressure (or pressure update), and a final velocity correction to obtain a divergence-free vector field. In some situations, the equations for the velocities are solved explicitly, so that the numerical most expensive step is the elliptic pressure problem. We here propose to solve this Poisson problem by a domain decomposition method which does not need any communication between the sub-regions. Hence, this system is perfectly adapted for parallel computation. We show under certain assumptions that this new scheme has the same order of convergence as the original pressure correction scheme (with global projection). Numerical examples for the Stokes system show the effectivity of this new pressure correction method. The convergence order O(k(2)) for resulting velocity fields can be observed in the norm l(2) (0, T; L-2 (Omega)).

This work surveys an r-adaptive moving mesh finite element method for the numerical solution of premixed laminar flame problems. Since the model of chemically reacting flow involves many different modes with diverse length scales, the computation of such a problem is often extremely time-consuming. Importantly, to capture the significant characteristics of the flame structure when using detailed chemistry, a much more stringent requirement on the spatial resolution of the interior layers of some intermediate species is necessary. Here, we propose a moving mesh method in which the mesh is obtained from the solution of so-called moving mesh partial differential equations. Such equations result from the variational formulation of a minimization problem for a given target functional that characterizes the inherent difficulty in the numerical approximation of the underlying physical equations. Adaptive mesh movement has emerged as an area of intense research in mesh adaptation in the last decade. With this approach points are only allowed to be shifted in space leaving the topology of the grid unchanged. In contrast to methods with local refinement, data structure hence is unchanged and load balancing is not an issue as grid points remain on the processor where they are. We will demonstrate the high potential of moving mesh methods for effectively optimizing the distribution of grid points to reach the required resolution for chemically reacting flows with extremely thin boundary layers.