Related papers: A new anisotropic mesh adaptation method based upo…
In this paper we study the loss of precision of numerical methods discretizing anisotropic problems and propose alternative approaches free from this drawback. The deterioration of the accuracy is observed when the coordinates and the mesh…
In this paper we develop an $hp$-adaptive procedure for the numerical solution of general, semilinear elliptic boundary value problems in 1d, with possible singular perturbations. Our approach combines both a prediction-type adaptive Newton…
We propose a new method for the construction of layer-adapted meshes for singularly perturbed differential equations (SPDEs), based on mesh partial differential equations (MPDEs) that incorporate \emph{a posteriori} solution information.…
An integro-differential equation of hyperbolic type, with mixed boundary conditions, is considered. A continuous space-time finite element method of degree one is formulated. A posteriori error representations based on space-time cells is…
In this paper we develop an adaptive procedure for the numerical solution of general, semilinear elliptic problems with possible singular perturbations. Our approach combines both a prediction-type adaptive Newton method and an adaptive…
A robust and fast solver for the fractional differential equation (FDEs) involving the Riesz fractional derivative is developed using an adaptive finite element method on non-uniform meshes. It is based on the utilization of hierarchical…
This paper studies the problem of estimating the means $\pm\theta_{*}\in\mathbb{R}^{d}$ of a symmetric two-component Gaussian mixture $\delta_{*}\cdot N(\theta_{*},I)+(1-\delta_{*})\cdot N(-\theta_{*},I)$ where the weights $\delta_{*}$ and…
Based on neural network and adaptive subspace approximation method, we propose a new machine learning method for solving partial differential equations. The neural network is adopted to build the basis of the finite dimensional subspace.…
In [Heimann, Lehrenfeld, Preu{\ss}, SIAM J. Sci. Comp. 45(2), 2023, B139 - B165] new geometrically unfitted space-time Finite Element methods for partial differential equations posed on moving domains of higher-order accuracy in space and…
We present an a posteriori error analysis for the mixed virtual element method (mixed VEM) applied to second order elliptic equations in divergence form with mixed boundary conditions. The resulting error estimator is of residual-type. It…
We consider the reliable implementation of high-order unfitted finite element methods on Cartesian meshes with hanging nodes for elliptic interface problems. We construct a reliable algorithm to merge small interface elements with their…
In this work, we propose and analyze a pointwise a posteriori error estimator for simple eigenvalues of elliptic eigenvalue problems with adaptive finite element methods (AFEMs). We prove the reliability and efficiency of the residual-type…
This paper builds on recent developments of adaptive methods for linear transport equations based on certain stable variational formulations of Petrov-Galerkin type. The variational formulations allow us to employ meshes with cells of…
We present a fast adaptive method for the evaluation of heat potentials, which plays a key role in the integral equation approach for the solution of the heat equation, especially in a non-stationary domain. The algorithm utilizes a…
We analyze an adaptive boundary element method for the weakly-singular and hypersingular integral equations for the 2D and 3D Helmholtz problem. The proposed adaptive algorithm is steered by a residual error estimator and does not rely on…
This work develops user-friendly a posteriori error estimates of finite element methods, based on smoothers of linear iterative solvers. The proposed method employs simple smoothers, such as Jacobi or Gauss-Seidel iteration, on an auxiliary…
For solving pseudo-convex global optimization problems, we present a novel fully adaptive steepest descent method (or ASDM) without any hard-to-estimate parameters. For the step-size regulation in an $\varepsilon$-normalized direction, we…
We present an anisotropic goal-oriented error estimator based on the Dual Weighted Residual (DWR) method for time-dependent convection-dominated problems. Using elementwise p-anisotropic finite element spaces, the estimator is elementwise…
We present precise anisotropic interpolation error estimates for smooth functions using a new geometric parameter and derive inverse inequalities on anisotropic meshes. In our theory, the interpolation error is bounded in terms of the…
The p-norm often used in stress constrained topology optimisation supposedly mimics a delta function and it is thus characterised by a small length scale and ideally one would also prefer to have the solid-void transition occur over a small…