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The prototypical diffuse-interface model for incompressible fluid mixtures is the Navier–Stokes Cahn–Hilliard (Allen–Cahn) model. Despite its foundation in continuum mixture theory, it is not fully compatible with this theory due to the diffusive flux approximation. This paper introduces a class of thermodynamically consistent diffuse-interface incompressible fluid mixture models that is fully compatible with the continuum theory of mixtures. The proposed models can be formulated in either constituent or mixture quantities, enabling a direct comparison with the Navier–Stokes Cahn–Hilliard (Allen–Cahn) model with non-matching densities. This comparison reveals the key modelling simplifications employed in the latter. [ABSTRACT FROM AUTHOR]

The travel time functional measures the time taken for a particle trajectory to travel from a given initial position to the boundary of the domain. Such evaluation is paramount in the postclosure safety assessment of deep geological storage facilities for radioactive waste where leaked, nonsorbing solutes can be transported to the surface of the site by the surrounding groundwater. The accurate simulation of this transport can be attained using standard dual-weighted-residual techniques to derive goal-oriented a posteriori error bounds. This work provides a key aspect in obtaining a suitable error estimate for the travel time functional: the evaluation of its Gateaux derivative. A mixed finite element method is implemented to approximate Darcy's equations, and numerical experiments are presented to test the performance of the proposed error estimator. In particular, we consider a test case inspired by the Sellafield site located in Cumbria, UK. [ABSTRACT FROM AUTHOR]

Recent developments in the context of minimum residual finite element methods are paving the way for designing quasi-optimal discretization methods in non-standard function spaces, such as q-type Sobolev spaces. For q → 1, these methods have demonstrated huge potential in avoiding the notorious Gibbs phenomena, i.e., the occurrence of spurious non-physical oscillations near thin layers and jump discontinuities. In this work we provide theoretical results that explain some of these numerical observations. In particular, we investigate the Gibbs phenomena for q-best approximations of discontinuities in finite element spaces with 1 ≤ q < ∞. We prove sufficient conditions on meshes in one and two dimensions such that over- and undershoots vanish in the limit q → 1. Moreover, we include examples of meshes such that Gibbs phenomena remain present even for q = 1 and demonstrate that our results can be used to design meshes so as to eliminate the Gibbs phenomenon. [ABSTRACT FROM AUTHOR]

This work presents a comprehensive discretization theory for abstract linear operator equations in Banach spaces. The fundamental starting point of the theory is the idea of residual minimization in dual norms and its inexact version using discrete dual norms. It is shown that this development, in the case of strictly convex reflexive Banach spaces with strictly convex dual, gives rise to a class of nonlinear Petrov-Galerkin methods and, equivalently, abstract mixed methods with monotone nonlinearity. Under the Fortin condition, we prove discrete stability and quasi-optimal convergence of the abstract inexact method, with constants depending on the geometry of the underlying Banach spaces. The theory generalizes and extends the classical Petrov-Galerkin method as well as existing residual-minimization approaches, such as the discontinuous Petrov-Galerkin method. [ABSTRACT FROM AUTHOR]

We propose and analyze a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting. The weak formulation allows for the direct approximation of solutions in the Lebesgue L p {L^{p}} -space, 1 < p < ∞ {1

While it is classical to consider the solution of the convection-diffusion-reaction equation in the Hilbert space H 0 1 ⁢ (Ω) {H_{0}^{1}(\Omega)} , the Banach Sobolev space W 0 1 , q ⁢ (Ω) {W^{1,q}_{0}(\Omega)} , 1 < q < ∞ {1

We present an adaptive methodology for the solution of (linear and) nonlinear time dependent problems that is especially tailored for massively parallel computations. The basic concept is to solve for large blocks of space-time unknowns instead of marching sequentially in time. The methodology is a combination of a computationally efficient implementation of a parallel-in-space-time finite element solver coupled with a posteriori space-time error estimates and a parallel mesh generator. While we focus on spatial adaptivity in this work, the methodology enables simultaneous adaptivity in both space and time domains. We explore this basic concept in the context of a variety of time steppers including Θ-schemes and backward difference formulas. We specifically illustrate this framework with applications involving time dependent linear, quasi-linear, and semilinear diffusion equations. We focus on investigating how the coupled space-time refinement indicators for this class of problems affect spatial adaptivity. Finally, we show good scaling behavior up to 150,000 processors on the NCSA Blue Waters machine. This conceptually simple methodology enables scaling on next generation multicore machines by simultaneously solving for a large number of timesteps, and reducing computational overhead by locally refining spatial blocks that can track localized features. This methodology also opens up the possibility of efficiently incorporating adjoint equations for error estimators and inverse design problems, since blocks of space-time are simultaneously solved and stored in memory. [ABSTRACT FROM AUTHOR]

SUMMARY In this paper, we develop a thermodynamically consistent four-species model of tumor growth on the basis of the continuum theory of mixtures. Unique to this model is the incorporation of nutrient within the mixture as opposed to being modeled with an auxiliary reaction-diffusion equation. The formulation involves systems of highly nonlinear partial differential equations of surface effects through diffuse-interface models. A mixed finite element spatial discretization is developed and implemented to provide numerical results demonstrating the range of solutions this model can produce. A time-stepping algorithm is then presented for this system, which is shown to be first order accurate and energy gradient stable. The results of an array of numerical experiments are presented, which demonstrate a wide range of solutions produced by various choices of model parameters.Copyright © 2011 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Abstract: A methodology for designing formally second-order time-accurate and yet loosely-coupled partitioned procedures for the solution of nonlinear fluid–structure interaction (FSI) problems on moving grids is presented. Its key components are a fluid time-integrator that is provably second-order time-accurate on moving grids, the midpoint rule for advancing in time the solution of the structural dynamics equations of motion, a second-order structure predictor for bypassing the inner-iterations encountered in strongly-coupled solution procedures, and a carefully designed algorithm for time-integrating the motion of the fluid-mesh. Following this methodology, two different loosely-coupled schemes are constructed for the solution of transient nonlinear FSI problems and proved to be second-order time-accurate. Three-dimensional numerical results pertaining to the simulation of the aeroelastic response to a gravity excitation of a complete F-16 configuration are also presented. In addition to confirming the theoretical results discussed in this paper, these numerical results highlight a very stable behavior of the designed loosely-coupled partitioned procedures. [Copyright &y& Elsevier]

We introduce the concept of machine-learning minimal-residual (ML-MRes) finite element discretizations of partial differential equations (PDEs), which resolve quantities of interest with striking accuracy, regardless of the underlying mesh size. The methods are obtained within a machine-learning framework during which the parameters defining the method are tuned against available training data. In particular, we use a provably stable parametric Petrov–Galerkin method that is equivalent to a minimal-residual formulation using a weighted norm. While the trial space is a standard finite element space, the test space has parameters that are tuned in an off-line stage. Finding the optimal test space therefore amounts to obtaining a goal-oriented discretization that is completely tailored towards the quantity of interest. We use an artificial neural network to define the parametric family of test spaces. Using numerical examples for the Laplacian and advection equation in one and two dimensions, we demonstrate that the ML-MRes finite element method has superior approximation of quantities of interest even on very coarse meshes. [ABSTRACT FROM AUTHOR]

In this article we consider the numerical approximation of the convection-diffusion-reaction equation. One of the main challenges of designing a numerical method for this problem is that boundary layers occurring in the convection-dominated case can lead to non-physical oscillations in the numerical approximation, often referred to as Gibbs phenomena. The idea of this article is to consider the approximation problem as a residual minimization in dual norms in L q -type Sobolev spaces, with 1 < q < ∞. We then apply a non-standard, non-linear Petrov–Galerkin discretization, that is applicable to reflexive Banach spaces such that the space itself and its dual are strictly convex. Similar to discontinuous Petrov–Galerkin methods, this method is based on minimizing the residual in a dual norm. Replacing the intractable dual norm by a suitable discrete dual norm gives rise to a non-linear inexact mixed method. This generalizes the Petrov–Galerkin framework developed in the context of discontinuous Petrov–Galerkin methods to more general Banach spaces. For the convection–diffusion–reaction equation, this yields a generalization of a similar approach from the L 2 -setting to the L q -setting. A key advantage of considering a more general Banach space setting is that, in certain cases, the oscillations in the numerical approximation vanish as q tends to 1, as we will demonstrate using a few simple numerical examples. [ABSTRACT FROM AUTHOR]