Related papers: Mesh Sensitivity Analysis for Finite Element Solut…
We construct a finite element method (FEM) for the infinity Laplacian. Solutions of this problem may be singular, which has prompted us to conduct an a posteriori analysis of the method deriving residual based estimators to drive an…
In this paper we establish existence, uniqueness, and boundedness results for an elliptic variational inequality coupled with a nonlinear ordinary differential equation. Under the general framework, we present a new application modelling…
We present a continuous finite element method for some examples of fully nonlinear elliptic equation. A key tool is the discretisation proposed in Lakkis & Pryer (2011, SISC) allowing us to work directly on the strong form of a linear PDE.…
In this paper, a direct finite element method is proposed for solving interface problems on unfitted meshes. This new method treats the two interface conditions as an $H^{\frac12}(\Gamma)\times H^{-\frac12}(\Gamma)$ pair for the mutual…
We formulate and analyze a goal-oriented adaptive finite element method for a symmetric linear elliptic partial differential equation (PDE) that can simultaneously deal with multiple linear goal functionals. In each step of the algorithm,…
A type of adaptive finite element method for the eigenvalue problems is proposed based on the multilevel correction scheme. In this method, adaptive finite element method to solve eigenvalue problems involves solving associated boundary…
In this paper we propose a finite element method for solving elliptic equations with the observational Dirichlet boundary data which may subject to random noises. The method is based on the weak formulation of Lagrangian multiplier. We show…
We describe and analyze a hybrid finite element/neural network method for predicting solutions of partial differential equations. The methodology is designed for obtaining fine scale fluctuations from neural networks in a local manner. The…
This article considers a model problem of elastoplasticity with linearly kinematic hardening and presents hp-finite element discretizations of two equivalent weak formulations each having their respective advantages. A mixed variational…
A new field of numerical astrophysics is introduced which addresses the solution of large, multidimensional structural or slowly-evolving problems (rotating stars, interacting binaries, thick advective accretion disks, four dimensional…
The Poisson-Boltzmann equation is a widely used model to study the electrostatics in molecular solvation. Its numerical solution using a boundary integral formulation requires a mesh on the molecular surface only, yielding accurate…
We consider adaptive finite element methods for solving a multiscale system consisting of a macroscale model comprising a system of reaction-diffusion partial differential equations coupled to a microscale model comprising a system of…
We study iterative finite element approximations for the numerical approximation of semilinear elliptic boundary value problems with monotone nonlinear reactions of subcritical growth. The focus of our contribution is on an optimal a priori…
In the context of unfitted finite element discretizations the realization of high order methods is challenging due to the fact that the geometry approximation has to be sufficiently accurate. Recently a new unfitted finite element method…
We develop a finite element method for elliptic partial differential equations on so called composite surfaces that are built up out of a finite number of surfaces with boundaries that fit together nicely in the sense that the intersection…
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…
We introduce a unified sensitivity concept for shape and topological perturbations and perform the sensitivity analysis for a discretized PDE-constrained design optimization problem in two space dimensions. We assume that the design is…
We design an adaptive unfitted finite element method on the Cartesian mesh with hanging nodes. We derive an hp-reliable and efficient residual type a posteriori error estimate on K-meshes. A key ingredient is a novel hp-domain inverse…
We consider linear systems arising from the use of the finite element method for solving scalar linear elliptic problems. Our main result is that these linear systems, which are symmetric and positive semidefinite, are well approximated by…
A simple and efficient interface-fitted mesh generation algorithm is developed in this paper. This algorithm can produce a local anisotropic fitting mixed mesh which consists of both triangles and quadrilaterals near the interface. A new…