Related papers: A Recovery-Based Error Indicator for Finite Differ…
This paper presents an a priori error analysis of the Deep Mixed Residual method (MIM) for solving high-order elliptic equations with non-homogeneous boundary conditions, including Dirichlet, Neumann, and Robin conditions. We examine MIM…
In this article, a posteriori error analysis of the elliptic obstacle problem is addressed using hybrid high-order methods. The method involve cell unknowns represented by degree-$r$ polynomials and face unknowns represented by degree-$s$…
In this paper, we propose a novel recovery based finite element method for the Cahn-Hilliard equation. One distinguishing feature of the method is that we discretize the fourth-order differential operator in a standard $C^0$ linear finite…
This paper presents a high-order method for solving an interface problem for the Poisson equation on embedded meshes through a coupled finite element and integral equation approach. The method is capable of handling homogeneous or…
This paper presents an innovative continuous linear finite element approach to effectively solve biharmonic problems on surfaces. The key idea behind this method lies in the strategic utilization of a surface gradient recovery operator to…
There have been extensive studies on solving differential equations using physics-informed neural networks. While this method has proven advantageous in many cases, a major criticism lies in its lack of analytical error bounds. Therefore,…
An error control technique aimed to assess the quality of smoothed finite element approximations is presented in this paper. Finite element techniques based on strain smoothing appeared in 2007 were shown to provide significant advantages…
We propose and analyze a two-level method for mimetic finite difference approximations of second order elliptic boundary value problems. We prove that the two-level algorithm is uniformly convergent, i.e., the number of iterations needed to…
Adaptive finite element methods are a powerful tool to obtain numerical simulation results in a reasonable time. Due to complex chemical and mechanical couplings in lithium-ion batteries, numerical simulations are very helpful to…
This work delves into solving the two dimensional Poisson problem through the Finite Element Method which is relevant in various physical scenarios including heat conduction, electrostatics, gravity potential, and fluid dynamics. However,…
This paper introduces a new recovery-based a posteriori error estimator for the lowest order Nedelec finite element approximation to the H(curl) interface problem. The error estimator is analyzed by establishing both the reliability and the…
We derive a residual-based $hp$-a posteriori error estimator for hybrid high-order (HHO) methods on simplicial meshes applied to the biharmonic problem posed on two- and three-dimensional polytopal Lipschitz domains. The a posteriori error…
We consider an optimal recovery problem for the Poisson problem when the boundary data is unknown. Compensating information is provided in the form of a finite number of measurements of the solution. A finite element algorithm for this…
We consider a mixed variational formulation recently proposed for the coupling of the Brinkman--Forchheimer and Darcy equations and develop the first reliable and efficient residual-based a posteriori error estimator for the 2D version of…
The estimation of distributed parameters in partial differential equations (PDE) from measures of the solution of the PDE may lead to under-determination problems. The choice of a parameterization is a usual way of adding a-priori…
Accurate error estimation is crucial in model order reduction, both to obtain small reduced-order models and to certify their accuracy when deployed in downstream applications such as digital twins. In existing a posteriori error estimation…
This work is about a new two-level solver for Helmholtz equations discretized by finite elements. The method is inspired by two-grid methods for finite-difference Helmholtz problems as well as by previous work on two-level…
A residual based {\em a posteriori} error estimator is derived for a quadratic finite element method (fem) for the elliptic obstacle problem. The error estimator involves various residuals consisting the data of the problem, discrete…
In this paper, we develop a Bayesian multiscale approach based on a multiscale finite element method. Because of scale disparity in many multiscale applications, computational models can not resolve all scales. Various subgrid models are…
We propose a polynomial preserving recovery method for PHT-splines within isogeometric analysis to obtain more accurate gradient approximations. The method fully exploits the local interpolation properties of PHT-splines and avoids the need…