Related papers: Analysis of adaptive two-grid finite element algor…
This work introduces a new method to efficiently solve optimization problems constrained by partial differential equations (PDEs) with uncertain coefficients. The method leverages two sources of inexactness that trade accuracy for speed:…
We prove convergence with optimal algebraic rates for an adaptive finite element method for nonlinear equations with strongly monotone operator. Unlike prior works, our analysis also includes the iterative and inexact solution of the…
Multiscale and multiphysics problems need novel numerical methods in order for them to be solved correctly and predictively. To that end, we develop a wavelet based technique to solve a coupled system of nonlinear partial differential…
We study dendritic microstructure evolution using an adaptive grid, finite element method applied to a phase-field model. The computational complexity of our algorithm, per unit time, scales linearly with system size, rather than the…
We propose an adaptive multigrid preconditioning technology for solving linear systems arising from Discontinuous Petrov-Galerkin (DPG) discretizations. Unlike standard multigrid techniques, this preconditioner involves only trace spaces…
Partial differential equations (PDEs) are typically used as models of physical processes but are also of great interest in PDE-based image processing. However, when it comes to their use in imaging, conventional numerical methods for…
This study presents constructions of the space-time Conservation Element and Solution Element (CESE) methods to accommodate adaptive unstructured quadrilateral meshes. Subsequently, a novel algorithm is devised to effectively manage the…
In this article, we propose a two-grid based adaptive proper orthogonal decomposition (POD) method to solve the time dependent partial differential equations. Based on the error obtained in the coarse grid, we propose an error indicator for…
A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz…
This work proposes an $r$-adaptive finite element method (FEM) using neural networks (NNs). The method employs the Ritz energy functional as the loss function, currently limiting its applicability to symmetric and coercive problems, such as…
This paper is concerned with the numerical approximation of quantities of interest associated with solutions to parametric elliptic partial differential equations (PDEs). The key novelty of this work is in its focus on the quantities of…
Many problems in fluid modelling require the efficient solution of highly anisotropic elliptic partial differential equations (PDEs) in "flat" domains. For example, in numerical weather- and climate-prediction an elliptic PDE for the…
New finite element methods are proposed for elliptic interface problems in one and two dimensions. The main motivation is not only to get an accurate solution but also an accurate first order derivative at the interface (from each side).…
For adaptive mixed finite element methods (AMFEM), we first introduce the data oscillation to analyze, without the restriction that the inverse of the coefficient matrix of the partial differential equations (PDEs) is a piecewise polynomial…
We consider a second-order elliptic boundary value problem with strongly monotone and Lipschitz-continuous nonlinearity. We design and study its adaptive numerical approximation interconnecting a finite element discretization, the…
We present the first rigorous convergence analysis of the smoothed adaptive finite element method (S-AFEM) proposed in [Mulita, Giani, Heltai: SIAM J. Sci. Comput. 43, 2021]. S-AFEM modifies the classical adaptive finite element method…
The convergence analysis for least-squares finite element methods led to various adaptive mesh-refinement strategies: Collective marking algorithms driven by the built-in a posteriori error estimator or an alternative explicit…
The paper aims to establish a fully discrete finite element (FE) scheme and provide cost-effective solutions for one-dimensional time-space Caputo-Riesz fractional diffusion equations on a bounded domain $\Omega$. Firstly, we construct a…
We introduce the concept of data-driven finite element methods. These are finite-element discretizations of partial differential equations (PDEs) that resolve quantities of interest with striking accuracy, regardless of the underlying mesh…
We propose a new paradigm for designing efficient p-adaptive arbitrary high order methods. We consider arbitrary high order iterative schemes that gain one order of accuracy at each iteration and we modify them in order to match the…