Related papers: On the best accuracy using the $h$-adaptive finite…
We consider an adaptive finite element method with arbitrary but fixed polynomial degree $p \ge 1$, where adaptivity is driven by an edge-based residual error estimator. Based on the modified maximum criterion from [Diening et al, Found.…
Adaptive Finite Element Method (adaptivity) is known to be an effective numerical tool for some ill-posed problems. The key advantage of the adaptivity is the image improvement with local mesh refinements. A rigorous proof of this property…
In this paper a new hp-adaptive strategy for elliptic problems based on refinement history is proposed, which chooses h-, p- or hp-refinement on individual elements according to a posteriori error estimate, as well as smoothness estimate of…
In this work we present an adaptive Newton-type method to solve nonlinear constrained optimization problems in which the constraint is a system of partial differential equations discretized by the finite element method. The adaptive…
This paper aims to develop an efficient adaptive finite element method for the second-order elliptic problem. Although the theory for adaptive finite element methods based on residual-type a posteriori error estimator and bisection…
In this work we develop and analyze an adaptive finite element method for efficiently solving electrical impedance tomography -- a severely ill-posed nonlinear inverse problem for recovering the conductivity from boundary voltage…
We investigate the application of a posteriori error estimates to a fractional optimal control problem with pointwise control constraints. Specifically, we address a problem in which the state equation is formulated as an integral form of…
Consider the diffraction of an electromagnetic plane wave by a biperiodic structure where the wave propagation is governed by the three-dimensional Maxwell equations. Based on transparent boundary condition, the grating problem is…
The overfitting is one of the cursing subjects in the deep learning field. To solve this challenge, many approaches were proposed to regularize the learning models. They add some hyper-parameters to the model to extend the generalization;…
Algorithms that promise to leverage resources of quantum computers efficiently to accelerate the finite element method have emerged. However, the finite element method is usually incorporated into a high-level numerical scheme which allows…
We consider second-order PDE problems set in unbounded domains and discretized by Lagrange finite elements on a finite mesh, thus introducing an artificial boundary in the discretization. Specifically, we consider the reaction diffusion…
In this work, an adaptive edge element method is developed for an H(curl)-elliptic constrained optimal control problem. We use the lowest-order Nedelec's edge elements of first family and the piecewise (element-wise) constant functions to…
The construction of finite element approximations in $\mathbf{H}(\mbox{div}, {\Omega})$ usually requires the Piola transformation to map vector polynomials from a master element to vector fields in the elements of a partition of the region…
This work introduces an adaptive mesh refinement technique for hierarchical hybrid grids with the goal to reach scalability and maintain excellent performance on massively parallel computer systems. On the block structured hierarchical…
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…
Using the standard finite element method (FEM) to solve general partial differential equations, the round-off error is found to be proportional to $N^{\beta_{\rm R}}$, with $N$ the number of degrees of freedom (DoFs) and $\beta_{\rm R}$ a…
Automatic algorithms attempt to provide approximate solutions that differ from exact solutions by no more than a user-specified error tolerance. This paper describes an automatic, adaptive algorithm for approximating the solution to a…
This article is concerned with the numerical solution of convex variational problems. More precisely, we develop an iterative minimisation technique which allows for the successive enrichment of an underlying discrete approximation space in…
Consider the scattering of an elastic plane wave by a rigid obstacle, which is immersed in a homogeneous and isotropic elastic medium in two dimensions. Based on a Dirichlet-to-Neumann (DtN) operator, an exact transparent boundary condition…
This paper reviews the state of the art and discusses very recent mathematical developments in the field of adaptive boundary element methods. This includes an overview of available a posteriori error estimates as well as a state-of-the-art…