Abstract. The very weak formulation of the porous medium/fast diffusion equation yields an evolution problem in a Gelfand triple with the pivot space H. This allows to employ methods of the theory of monotone operators in order to study fully discrete approximations combining a Galerkin method (including conforming finite element methods) with the backward Euler scheme. Convergence is shown eve...

A survey is given of numerical methods for calculating fixed points of nonlinear integral operators. The emphasis is on general methods, ones that are applicable to a wide variety of nonlinear integral equations. These methods include projection methods (Galerkin and collocation) and Nyström methods. Some of the practical problems related to the implementation of these methods is also discussed...

A general algorithm is developed that reuses available information to accelerate the iterative convergence of linear systems with multiple right-hand sides A x = b i, which are commonly encountered in steady or unsteady simulations of nonlinear equations. The algorithm is based on the classical GMRES algorithm with eigenvector enrichment but also includes a Galerkin projection preprocessing ste...

In this paper, we consider the issue of convergence toward entropy solutions for high order finite volume weighted essentially non-oscillatory (WENO) scheme and discontinuous Galerkin (DG) finite element method approximating scalar nonconvex conservation laws. Although such high order nonlinearly stable schemes can usually converge to entropy solutions of convex conservation laws, convergence m...

A Petrov-Galerkin discretization is studied of an ultra-weak variational formulation of the convection-diffusion equation in mixed form. To arrive at an implementable method, the truly optimal test space has to be replaced by its projection onto a finite dimensional test search space. To prevent that this latter space has to be taken increasingly large for vanishing diffusion, a formulation is ...

This paper is concerned with the numerical solution of large scale Sylvester equations AX −XB = C, Lyapunov equations as a special case in particular included, with C having very small rank. For stable Lyapunov equations, Penzl (2000) and Li and White (2002) demonstrated that the so called Cholesky factor ADI method with decent shift parameters can be very effective. In this paper we present a ...

We consider finite element projection methods for linear partial differential equations, in which the spaces of trial functions and test functions may be different. In addition to approximation and smoothness properties, conditions implying equality of dimensions and uniform coerciveness are required, the most important of which resembles a strong form of an inverse assumption. Our results prov...

We present a discontinuous Galerkin method on a fully unstructured grid for the modeling of unsteady incompressible fluid flows with free surfaces. The surface is modeled by embedding and represented by a level set. We discuss the discretization of the flow equations and the level set equation as well a various ways of advancing the equations in time using velocity projection techniques. The ef...

A reduced basis method based on a physics-informed machine learning framework is developed for efficient reduced-order modeling of parametrized partial differential equations (PDEs). feedforward neural network used to approximate the mapping from time-parameter coefficients. During offline stage, trained by minimizing weighted sum residual loss equations, and data labeled coefficients that are ...

Abstract In this work, we develop projection kernels for Euler-Lagrange point-particle simulations of disperse multiphase flows on arbitrary curved elements. These are employed in a high-order discontinuous Galerkin framework projecting the action particles to Eulerian mesh. Instead commonly used isotropic kernels, such as Gaussian-type kernel, construct an anisotropic polynomial-based smoothin...