Related papers: A Cut Finite Element Method with Boundary Value Co…
A finite element (FE) discretization for the steady, incompressible, fully inhomogeneous, generalized Navier-Stokes equations is proposed. By the method of divergence reconstruction operators, the formulation is valid for all shear stress…
We study cut finite element discretizations of a Darcy interface problem based on the mixed finite element pairs $\textbf{RT}_k\times Q_k$, $k\geq 0$. Here $Q_k$ is the space of discontinuous polynomial functions of degree less or equal to…
This article deals with error estimates for the finite element approximation of variational normal derivatives and, as a consequence, error estimates for the finite element approximation of Dirichlet boundary control problems with energy…
In recent papers the author introduced a simple alternative to isoparametric finite elements of the n-simplex type, to enhance the accuracy of approximations of second-order boundary value problems with Dirichlet conditions, posed in smooth…
We prove strong convergence for a large class of finite element methods for the time-dependent Joule heating problem in three spatial dimensions with mixed boundary conditions on Lipschitz domains. We consider conforming subspaces for the…
Obtaining high-precision guaranteed lower eigenvalue bounds remains difficult, even though the standard high-order conforming finite element (FEM) easily yields extremely sharp upper bounds. Recently developed rigorous approaches using such…
An important step in shape optimization with partial differential equation constraints is to adapt the geometry during each optimization iteration. Common strategies are to employ mesh-deformation or re-meshing, where one or the other…
We consider a unique continuation problem where the Dirichlet trace of the solution is known to have finite dimension. We prove Lipschitz stability of the unique continuation problem and design a finite element method that exploits the…
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 nonconforming finite element method for the solution of the ill-posed elliptic Cauchy problem. We prove error estimates using continuous dependence estimates in the $L^2$-norm. The effect of perturbations in data…
A simple third order compact finite element method is proposed for one-dimensional Sturm-Liouville boundary value problems. The key idea is based on the interpolation error estimate, which can be related to the source term. Thus, a simple…
In many situations with finite element discretizations it is desirable or necessary to impose boundary or interface conditions not as essential conditions -- i.e. through the finite element space -- but through the variational formulation.…
This paper introduces an approach to decoupling singularly perturbed boundary value problems for fourth-order ordinary differential equations that feature a small positive parameter $\epsilon$ multiplying the highest derivative. We…
We present a stabilized finite element method for the numerical solution of cavitation in lubrication, modeled as an inequality-constrained Reynolds equation. The cavitation model is written as a variable coefficient saddle-point problem…
Nitsche's method is a standard device for weakly imposing Dirichlet boundary conditions, but for the stabilized nonsymmetric formulation the available $L^2$-error analysis for Poisson's equation still predicts a half-order loss, whereas…
We develop a stabilized cut finite element method for the stationary convection diffusion problem on a surface embedded in ${\mathbb{R}}^d$. The cut finite element method is based on using an embedding of the surface into a three…
We develop a theoretical framework for the analysis of stabilized cut finite element methods for the Laplace-Beltrami operator on a manifold embedded in $\mathbb{R}^d$ of arbitrary codimension. The method is based on using continuous…
In this paper, we propose an unfitted finite element method to solve PDE-constrained shape optimization problems via shape gradient flow. The shape gradient flow system consists of the state equation, the adjoint equation, the velocity…
We present a domain decomposition strategy for developing structure-preserving finite element discretizations from data when exact governing equations are unknown. On subdomains, trainable Whitney form elements are used to identify…
We use a numerical-analytic technique to construct a sequence of successive approximations to the solution of a system of fractional differential equations, subject to Dirichlet boundary conditions. We prove the uniform convergence of the…