Related papers: A direct finite element method for elliptic interf…
When solving elliptic partial differential equations in a region containing immersed interfaces (possibly evolving in time), it is often desirable to approximate the problem using an independent background discretisation, not aligned with…
The aim of this paper is to propose an efficient adaptive finite element method for eigenvalue problems based on the multilevel correction scheme and inverse power method. This method involves solving associated boundary value problems on…
We study unique continuation over an interface using a stabilized unfitted finite element method tailored to the conditional stability of the problem. The interface is approximated using an isoparametric transformation of the background…
With the development of multi-layer elastic systems in the field of engineering mechanics, the corresponding variational inequality theory and algorithm design have received more attention and research. In this study, a class of equivalent…
A full multigrid finite element method is proposed for semilinear elliptic equations. The main idea is to transform the solution of the semilinear problem into a series of solutions of the corresponding linear boundary value problems on the…
Immersed finite element (IFE) methods are a group of long-existing numerical methods for solving interface problems on unfitted meshes. A core argument of the methods is to avoid mesh regeneration procedure when solving moving interface…
In this paper we propose a new finite element method for solving elliptic optimal control problems with pointwise state constraints, including the distributed controls and the Dirichlet or Neumann boundary controls. The main idea is to use…
We present a unified framework for developing and analysing immersed finite element (IFE) spaces for solving typical elliptic interface problems with interface independent meshes. This framework allows us to construct a group of new IFE…
We consider a randomised implementation of the finite element method (FEM) for elliptic partial differential equations on high-dimensional models. This is motivated by applications where model predictions are essential for real-time process…
We study the problem $-\Delta u = \gamma$, where $\gamma$ is a singular measure, with support on a curve or a point. We prove that optimal rates of convergence for the finite element method can be obtained using properly graded meshes. In…
A homogenization approach is one of effective strategies to solve multiscale elliptic problems approximately. The finite element heterogeneous multiscale method (FEHMM) which is based on the finite element makes possible to simulate such…
A new immersed finite element (IFE) method is developed for second-order elliptic problems with discontinuous diffusion coefficient. The IFE space is constructed based on the rotated Q1 nonconforming finite elements with the integral-value…
We propose an energy-stable parametric finite element method (ES-PFEM) for simulating solid-state dewetting of thin films in two dimensions via a sharp-interface model, which is governed by surface diffusion and contact line (point)…
In this paper, we propose a novel mesh-free numerical method for solving the elliptic interface problems based on deep learning. We approximate the solution by the neural networks and, since the solution may change dramatically across the…
A finite element method is introduced to track interface evolution governed by the level set equation. The method solves for the level set indicator function in a narrow band around the interface. An extension procedure, which is essential…
We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for…
Maxwell interface problems are of great importance in many electromagnetic applications. Unfitted mesh methods are especially attractive in 3D computation as they can circumvent generating complex 3D interface-fitted meshes. However, many…
A novel finite element method for the approximation of Maxwell's equations over hybrid two-dimensional grids is studied. The choice of appropriate basis functions and numerical quadrature leads to diagonal mass matrices which allow for…
This paper develops an enhanced finite element method for approximating a class of variational problems which exhibit the \textit{Lavrentiev gap phenomenon} in the sense that the minimum values of the energy functional have a nontrivial gap…
In this paper, we propose a novel unfitted finite element method for the simulation of multiple body contact. The computational mesh is generated independently of the geometry of the interacting solids, which can be arbitrarily complex. The…