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This paper introduces a numerical approach to solve singularly perturbed convection diffusion boundary value problems for second-order ordinary differential equations that feature a small positive parameter {\epsilon} multiplying the highest derivative. We specifically examine Dirichlet boundary conditions. To solve this differential equation, we propose an upwind finite difference method and incorporate the Shishkin mesh scheme to capture the solution near boundary layers. Our solver is both direct and of high accuracy, with computation time that scales linearly with the number of grid points. MATLAB code of the numerical recipe is made publicly available. We present numerical results to validate the theoretical results and assess the accuracy of our method. The tables and graphs included in this paper demonstrate the numerical outcomes, which indicate that our proposed method offers a highly accurate approximation of the exact solution.

In this paper, we investigate a novel monolithic algebraic multigrid solver for the discrete Stokes problem discretized with stable mixed finite elements. The algorithm is based on the use of the low-order $\pmb{\mathbb{P}}_1 \text{iso}\kern1pt\pmb{ \mathbb{P}}_2/ \mathbb{P}_1$ discretization as a preconditioner for a higher-order discretization, such as $\pmb{\mathbb{P}}_2/\mathbb{P}_1$. Smoothed aggregation algebraic multigrid is used to construct independent coarsenings of the velocity and pressure fields for the low-order discretization, resulting in a purely algebraic preconditioner for the high-order discretization (i.e., using no geometric information). Furthermore, we incorporate a novel block LU factorization technique for Vanka patches, which balances computational efficiency with lower storage requirements. The effectiveness of the new method is verified for the $\pmb{\mathbb{P}}_2/\mathbb{P}_1$ (Taylor-Hood) discretization in two and three dimensions on both structured and unstructured meshes. Similarly, the approach is shown to be effective when applied to the $\pmb{\mathbb{P}}_2/\mathbb{P}_1^{disc}$ (Scott-Vogelius) discretization on 2D barycentrically refined meshes. This novel monolithic algebraic multigrid solver not only meets but frequently surpasses the performance of inexact Uzawa preconditioners, demonstrating the versatility and robust performance across a diverse spectrum of problem sets, even where inexact Uzawa preconditioners struggle to converge.

We study step-wise time approximations of non-linear hyperbolic initial value problems. The technique used here is a generalization of the minimizing movements method, using two time-scales: one for velocity, the other (potentially much larger) for acceleration. The main applications are from elastodynamics namely so-called generalized solids. The evolution follows an underlying variational structure exploited by step-wise minimisation. We show for a large family of (elastic) energies that the introduced scheme is stable; allowing for non-linearities of highest order. If the highest order can assumed to be linear, we show that the limit solutions are regular and that the minimizing movements scheme converges with optimal linear rate. Thus this work extends numerical time-step minimization methods to the realm of hyperbolic problems.

In this paper, two novel classes of implicit exponential Runge-Kutta (ERK) methods are studied for solving highly oscillatory systems. Firstly, we analyze the symplectic conditions for two kinds of exponential integrators and obtain the symplectic method. In order to effectively solve highly oscillatory problems, we try to design the highly accurate implicit ERK integrators. By comparing the Taylor series of numerical solution with exact solution, it can be verified that the order conditions of two new kinds of exponential methods are identical to classical Runge-Kutta (RK) methods, which implies that using the coefficients of RK methods, some highly accurate numerical methods are directly formulated. Furthermore, we also investigate the linear stability regions for these exponential methods. Finally, numerical results not only display the long time energy preservation of the symplectic method, but also illustrate the accuracy and efficiency of these formulated methods in comparison with standard ERK methods.

In this paper, we conduct an in-depth investigation of the structural intricacies inherent to the Invariant Energy Quadratization (IEQ) method as applied to gradient flows, and we dissect the mechanisms that enable this method to uphold linearity and the conservation of energy simultaneously. Building upon this foundation, we propose two methods: Invariant Energy Convexification and Invariant Energy Functionalization. These approaches can be perceived as natural extensions of the IEQ method. Employing our novel approaches, we reformulate the system connected to gradient flow, construct a semi-discretized numerical scheme, and obtain a commensurate modified energy dissipation law for both proposed methods. Finally, to underscore their practical utility, we provide numerical evidence demonstrating these methods' accuracy, stability, and effectiveness when applied to both Allen-Cahn and Cahn-Hilliard equations.

This paper introduces an approach to decoupling singularly perturbed boundary value problems for fourth-order ordinary differential equations that feature a small positive parameter $\epsilon$ multiplying the highest derivative. We specifically examine Lidstone boundary conditions and demonstrate how to break down fourth-order differential equations into a system of second-order problems, with one lacking the parameter and the other featuring $\epsilon$ multiplying the highest derivative. To solve this system, we propose a mixed finite element algorithm and incorporate the Shishkin mesh scheme to capture the solution near boundary layers. Our solver is both direct and of high accuracy, with computation time that scales linearly with the number of grid points. We present numerical results to validate the theoretical results and the accuracy of our method.

The multigrid V-cycle method is a popular method for solving systems of linear equations. It computes an approximate solution by using smoothing on fine levels and solving a system of linear equations on the coarsest level. Solving on the coarsest level depends on the size and difficulty of the problem. If the size permits, it is typical to use a direct method based on LU or Cholesky decomposition. In the settings with large coarsest-level problems approximate solvers such as iterative Krylov subspace methods, or direct methods based on low-rank approximation, are often used. The accuracy of the coarsest-level solver is typically determined based on the experience of the users with the concrete problems and methods. In this paper we present an approach to analyzing the effects of approximate coarsest-level solves on the convergence of the V-cycle method for symmetric positive definite problems. Using this approach we discuss how the convergence of the V-cycle method may be affected by (1) the choice of the tolerance in a stopping criterion based on the relative residual norm for an iterative coarsest-level solver or (2) by the choices of the low-rank threshold parameter and finite precision arithmetic for a block low-rank direct coarsest-level solver.Furthermore we present new coarsest-level stopping criteria tailored to the multigrid method and suggest a heuristic strategy for their effective use in practical computations.

Finite element methods and kinematically coupled schemes that decouple the fluid velocity and structure's displacement have been extensively studied for incompressible fluid-structure interaction (FSI) over the past decade. While these methods are known to be stable and easy to implement, optimal error analysis has remained challenging. Previous work has primarily relied on the classical elliptic projection technique, which is only suitable for parabolic problems and does not lead to optimal convergence of numerical solutions to the FSI problems in the standard $L^2$ norm. In this article, we propose a new kinematically coupled scheme for incompressible FSI thin-structure model and establish a new framework for the numerical analysis of FSI problems in terms of a newly introduced coupled non-stationary Ritz projection, which allows us to prove the optimal-order convergence of the proposed method in the $L^2$ norm. The methodology presented in this article is also applicable to numerous other FSI models and serves as a fundamental tool for advancing research in this field.

We consider the problem of solving linear least squares problems in a framework where only evaluations of the linear map are possible. We derive randomized methods that do not need any other matrix operations than forward evaluations, especially no evaluation of the adjoint map is needed. Our method is motivated by the simple observation that one can get an unbiased estimate of the application of the adjoint. We show convergence of the method and then derive a more efficient method that uses an exact linesearch. This method, called random descent, resembles known methods in other context and has the randomized coordinate descent method as special case. We provide convergence analysis of the random descent method emphasizing the dependence on the underlying distribution of the random vectors. Furthermore we investigate the applicability of the method in the context of ill-posed inverse problems and show that the method can have beneficial properties when the unknown solution is rough. We illustrate the theoretical findings in numerical examples. One particular result is that the random descent method actually outperforms established transposed-free methods (TFQMR and CGS) in examples.

Constraint Programming (CP) is a declarative programming paradigm that allows for modeling and solving combinatorial optimization problems, such as the Job-Shop Scheduling Problem (JSSP). While CP solvers manage to find optimal or near-optimal solutions for small instances, they do not scale well to large ones, i.e., they require long computation times or yield low-quality solutions. Therefore, real-world scheduling applications often resort to fast, handcrafted, priority-based dispatching heuristics to find a good initial solution and then refine it using optimization methods. This paper proposes a novel end-to-end approach to solving scheduling problems by means of CP and Reinforcement Learning (RL). In contrast to previous RL methods, tailored for a given problem by including procedural simulation algorithms, complex feature engineering, or handcrafted reward functions, our neural-network architecture and training algorithm merely require a generic CP encoding of some scheduling problem along with a set of small instances. Our approach leverages existing CP solvers to train an agent learning a Priority Dispatching Rule (PDR) that generalizes well to large instances, even from separate datasets. We evaluate our method on seven JSSP datasets from the literature, showing its ability to find higher-quality solutions for very large instances than obtained by static PDRs and by a CP solver within the same time limit.