Numerical homogenization for nonlinear strongly monotone problems

In this work we introduce and analyze a new multiscale method for strongly nonlinear monotone equations in the spirit of the Localized Orthogonal Decomposition. A problem-adapted multiscale space is constructed by solving linear local fine-scale problems which is then used in a generalized finite element method. The linearity of the fine-scale problems allows their localization and, moreover, makes the method very efficient to use. The new method gives optimal a priori error estimates up to linearization errors beyond periodicity and scale separation and without assuming higher regularity of the solution. The effect of different linearization strategies is discussed in theory and practice. Several numerical examples including stationary Richards equation confirm the theory and underline the applicability of the method

Numerical homogenization for nonlinear strongly monotone problems

In this work we introduce and analyze a new multiscale method for strongly nonlinear monotone equations in the spirit of the Localized Orthogonal Decomposition. A problem-adapted multiscale space is constructed by solving linear local fine-scale problems which is then used in a generalized finite element method. The linearity of the fine-scale problems allows their localization and, moreover, makes the method very efficient to use. The new method gives optimal a priori error estimates up to linearization errors. The results neither require structural assumptions on the coefficient such as periodicity or scale separation nor higher regularity of the solution. The effect of different linearization strategies is discussed in theory and practice. Several numerical examples including stationary Richards equation confirm the theory and underline the applicability of the method

Numerical methods for time-fractional evolution equations with nonsmooth data: a concise overview

Over the past few decades, there has been substantial interest in evolution equations that involving a fractional-order derivative of order $\alpha\in(0,1)$ in time, due to their many successful applications in engineering, physics, biology and finance. Thus, it is of paramount importance to develop and to analyze efficient and accurate numerical methods for reliably simulating such models, and the literature on the topic is vast and fast growing. The present paper gives a concise overview on numerical schemes for the subdiffusion model with nonsmooth problem data, which are important for the numerical analysis of many problems arising in optimal control, inverse problems and stochastic analysis. We focus on the following aspects of the subdiffusion model: regularity theory, Galerkin finite element discretization in space, time-stepping schemes (including convolution quadrature and L1 type schemes), and space-time variational formulations, and compare the results with that for standard parabolic problems. Further, these aspects are showcased with illustrative numerical experiments and complemented with perspectives and pointers to relevant literature.Comment: 24 pages, 3 figure

New Discretization Methods for the Numerical Approximation of PDEs

The construction and mathematical analysis of numerical methods for PDEs is a fundamental area of modern applied mathematics. Among the various techniques that have been proposed in the past, some – in particular, finite element methods, – have been exceptionally successful in a range of applications. There are however a number of important challenges that remain, including the optimal adaptive finite element approximation of solutions to transport-dominated diffusion problems, the efficient numerical approximation of parametrized families of PDEs, and the efficient numerical approximation of high-dimensional partial differential equations (that arise from stochastic analysis and statistical physics, for example, in the form of a backward Kolmogorov equation, which, unlike its formal adjoint, the forward Kolmogorov equation, is not in divergence form, and therefore not directly amenable to finite element approximation, even when the spatial dimension is low). In recent years several original and conceptionally new ideas have emerged in order to tackle these open problems. The goal of this workshop was to discuss and compare a number of novel approaches, to study their potential and applicability, and to formulate the strategic goals and directions of research in this field for the next five years

Robust Numerical Methods for Singularly Perturbed Differential Equations--Supplements

The second edition of the book "Roos, Stynes, Tobiska -- Robust Numerical Methods for Singularly Perturbed Differential Equations" appeared many years ago and was for many years a reliable guide into the world of numerical methods for singularly perturbed problems. Since then many new results came into the game, we present some selected ones and the related sources.Comment: arXiv admin note: text overlap with arXiv:1909.0827

Existence and Uniqueness of Non-linear, Possibly Degenerate Parabolic PDEs, with Applications to Flow in Porous Media

In the branch of mathematical analysis known as functional analysis, one mainly studies functions defined on vector spaces. For partial differential equations (PDEs), this analysis has proven to be a mighty resource of understanding and modelling the behavior of the equations. Throughout this thesis, the work will focus of theory of function spaces and existence and uniqueness theorems for variational formulations in normed vector spaces. We will recast PDEs as variational problems with operators acting on normed spaces, and further seek to prove the existence and uniqueness of a solution by assigning certain properties to the operator. The outline of this thesis is as follows: In Chapter 1, we summarize the Basic Notions of Functional Analysis relevant for the later work in the thesis. We define operators, discuss monotonicity, present the theory of Sobolev spaces, and illustrate the finite element method, giving short hints to the future relevancy of the described properties. Linear Problems have been extensively studied in the past. In Chapter 2, we present three important theorems illustrating the conditions for existence and uniqueness of solutions for variational formulations of the type: (i) Galerkin formulations in Hilbert spaces: The Lax-Milgram Theorem, (ii) Petrov-Galerkin formulations in Hilbert spaces: The Babuška-Lax-Milgram Theorem, (iii) Petrov-Galerkin formulations in Banach spaces: The Banach-Nečas-Babuška Theorem, and give their proofs. Chapter 3 is dedicated to the study of Non-linear Problems. We seek to extend the ideas of the previous chapter to variational formulations containing a non-linearity b(·) depending on the solution we seek. This has a major application in the analysis of non-linear PDEs, which in general may not possess analytical solutions. To attack these types of problems, we define a weak formulation of the main problem, and discretize the domain of where a solution is sought. Next, existence and uniqueness is established through fixed point theorems, which will be given with proof. We will focus our study on two central problems: The Richards equation (a non-linear, possibly degenerate parabolic PDE) and a transport equation modelling reactive flow in porous media (two coupled PDEs). For the fully discrete (non-linear) formulation of Richards equation we show results for (i) a Lipschitz continuous non-linearity. Here we consider three cases: 3(a) First, a linearization scheme is proposed. We prove existence and uniqueness by using the Lax-Milgram Theorem in combination with the Banach Fixed Point Theorem. (b) Second, we make the assumption that the non-linearity is strongly monotone. Here, existence is proven by the Brouwer Fixed Point Theorem (c) Third, we let the non-linearity be monotone and add a regularization term to the fully discrete formulation. Here, we prove existence as in the previous step, and lastly show convergence of the regularized scheme to the fully discrete scheme. (ii) a Hölder continuous non-linearity. We give two results: (a) First, we prove existence for a monotone and bounded non-linearity. (b) Second, we state the result of existence for a strongly monotone non-linearity by the Brouwer Fixed Point Theorem. In the applications of Brouwer Fixed Point Theorem, the uniqueness of the problem is proved by assuming there exists two solutions and obtaining a contradiction through inequalities by showing estimates that can not be true. Lastly, in Chapter 4, a mathematical model of Two-phase Flow in porous media is studied. We discuss the case of a Lipschitz continuous saturation, and show for the first time a proof of existence and uniqueness of a solution for the fully discrete (non-linear) scheme, assuming the saturation to be Hölder continuous and strongly monotonically increasing. This is done by creating a regularization of the fully discrete scheme, further proving existence with the Brouwer Fixed Point Theorem, and finally showing convergence with the help of an a priori estimate.Masteroppgave i anvendt og beregningsorientert matematikkMAMN-MABMAB39

A priori error estimates for finite element methods with numerical quadrature for nonmonotone nonlinear elliptic problems

The effect of numerical quadrature in finite element methods for solving quasilinear elliptic problems of nonmonotone type is studied. Under similar assumption on the quadrature formula as for linear problems, optimal error estimates in the L 2 and the H 1 norms are proved. The numerical solution obtained from the finite element method with quadrature formula is shown to be unique for a sufficiently fine mesh. The analysis is valid for both simplicial and rectangular finite elements of arbitrary order. Numerical experiments corroborate the theoretical convergence rate

Operator splitting methods for systems of convection-diffusion equations: nonlinear error mechanisms and correction strategies

Authors final draft post-refereeing.Many numerical methods for systems of convection-diffusion equations are based upon an operator splitting formulation, where convective and diffusive forces are accounted for in separate substeps. We describe the nonlinear mechanism of the splitting error in such numerical methods in the onedimensional case, a mechanism that is intimately linked to the local linearizations introduced implicitly in the (hyperbolic) convection steps by the use of an entropy condition. For convection-dominated flows, we demonstrate that operator splitting methods typically generate a numerical widening of viscous fronts, unless the splitting step is of the same magnitude as the diffusion scale. To compensate for the potentially damaging splitting error, we propose a corrected operator splitting (COS) method for general systems of convection-diffusion equations with the ability of correctly resolving the nonlinear balance between the convective and diffusive forces. In particular, COS produces viscous shocks with correct structure also when the splitting step is large. A front tracking method for systems of conservation laws, which in turn relies heavily on a Riemann solver, constitutes an important part of our COS strategy. The proposed COS method is successfully applied to a system modelling two-phase, multicomponent flow in porous media and a triangular system modelling three-phase flow