Related papers: Improved $L^2$ and $H^1$ error estimates for the H…

In this paper we establish a stability barrier of a class of high-order Hermite-type discretization of 1D advection equations underlying the hybrid-variable (HV) and active flux (AF) methods. These methods seek numerical approximations to…

In this manuscript, we present a mixed finite element discretization for a class of boundary-damped anisotropic port-Hamiltonian systems. Using a multiplier method, we demonstrate that the resulting approximation model uniformly preserves…

This article is a review on basic concepts and tools devoted to a posteriori error estimation for problems solved with the Finite Element Method. For the sake of simplicity and clarity, we mostly focus on linear elliptic diffusion problems,…

Using the gradient discretisation method (GDM), we provide a complete and unified numerical analysis for non-linear variational inequalities (VIs) based on Leray--Lions operators and subject to non-homogeneous Dirichlet and Signorini…

Calibration of stochastic local volatility (SLV) models to their underlying local volatility model is often performed by numerically solving a two-dimensional non-linear forward Kolmogorov equation. We propose a novel finite volume (FV)…

In the present work, we investigate the computational efficiency afforded by higher-order finite-element discretization of the saddle-point formulation of orbital-free density functional theory. We first investigate the robustness of viable…

We present a novel approach for high-order accurate numerical differentiation on unstructured meshes of quadrilateral elements. To differentiate a given function, an auxiliary function with greater smoothness properties is defined which…

This article aims to establish a semi-analytical approach based on the homotopy perturbation method (HPM) to find the closed form or approximated solutions for the population balance equations such as Smoluchowski's coagulation,…

We develop and analyse finite volume methods for the Poisson problem with boundary conditions involving oblique derivatives. We design a generic framework, for finite volume discretisations of such models, in which internal fluxes are not…

We present a novel probabilistic finite element method (FEM) for the solution and uncertainty quantification of elliptic partial differential equations based on random meshes, which we call random mesh FEM (RM-FEM). Our methodology allows…

In this paper, elliptic control problems with pointwise box constraints on the state is considered, where the corresponding Lagrange multipliers in general only represent regular Borel measure functions. To tackle this difficulty, the…

We develop all of the components needed to construct an adaptive finite element code that can be used to approximate fractional partial differential equations, on non-trivial domains in $d\geq 1$ dimensions. Our main approach consists of…

We present and analyze a linearized finite element method (FEM) for the dynamical incompressible magnetohydrodynamics (MHD) equations. The finite element approximation is based on mixed conforming elements, where Taylor--Hood type elements…

We study a family of $H^m$-conforming piecewise polynomials based on artificial neural network, named as the finite neuron method (FNM), for numerical solution of $2m$-th order partial differential equations in $\mathbb{R}^d$ for any $m,d…

A general framework for goal-oriented a posteriori error estimation for finite volume methods is presented. The framework does not rely on recasting finite volume methods as special cases of finite element methods, but instead directly…

Present day electromagnetic field calculations have limitations that are due to techniques employing edge-based discretization methods. While these vector finite element methods solve the issues of tangential continuity of fields and the…

This paper investigates the convergence properties of the hypergradient descent method (HDM), a 25-year-old heuristic originally proposed for adaptive stepsize selection in stochastic first-order methods. We provide the first rigorous…

In this work, we show the convergence of adaptive lowest-order FEM (AFEM) for an elliptic obstacle problem with non-homogeneous Dirichlet data, where the obstacle \chi\ is restricted only by \chi\ in H^2(\Omega). The adaptive loop is…

We prove the quasi-optimal convergence of a standard adaptive finite element method (AFEM) for nonlinear elliptic second-order equations of monotone type. The adaptive algorithm is based on residual-type a posteriori error estimators and…

In this paper, we derive improved a priori error estimates for families of hybridizable interior penalty discontinuous Galerkin (H-IP) methods using a variable penalty for second-order elliptic problems. The strategy is to use a…