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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…
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…
This paper concerns the inclusion of Newton's method into an adaptive finite element method (FEM) for the solution of nonlinear partial differential equations (PDEs). It features an adaptive choice of the damping parameter in the Newton…
This article is concerned with the mathematical analysis of a family of adaptive importance sampling algorithms applied to diffusion processes. These methods, referred to as Adaptive Biasing Potential methods, are designed to efficiently…
We study the weak finite element method solving convection-diffusion equations. A weak finite element scheme is presented based on a spacial variational form. We established a weak embedding inequality that is very useful in the weak finite…
This article investigates residual a posteriori error estimates and adaptive mesh refinements for time-dependent boundary element methods for the wave equation. We obtain reliable estimates for Dirichlet and acoustic boundary conditions…
In this paper, we develop an adaptive finite element method for the nonlinear steady-state Poisson-Nernst-Planck equations, where the spatial adaptivity for geometrical singularities and boundary layer effects are mainly considered. As a…
A probabilistic approach is developed for the exact solution $u$ to a determinist partial differential equation as well as for its associated approximation $u^{(k)}_{h}$ performed by $P_k$ Lagrange finite element. Two limitations motivated…
In this paper we formulate and analyse adaptive (space-time) least-squares finite element methods for the solution of convection-diffusion equations. The convective derivative $\mathbf{v} \cdot \nabla u$ is considered as part of the total…
This is a study of certain finite element methods designed for convection-dominated, time-dependent partial differential equations. Specifically, we analyze high order space-time tensor product finite element discretizations, used in a…
We consider a saddle point formulation for a sixth order partial differential equation and its finite element approximation, for two sets of boundary conditions. We follow the Ciarlet-Raviart formulation for the biharmonic problem to…
A distributed adaptive algorithm is proposed to solve a node-specific parameter estimation problem where nodes are interested in estimating parameters of local interest, parameters of common interest to a subset of nodes and parameters of…
Presence of a high-dimensional stochastic parameter space with discontinuities poses major computational challenges in analyzing and quantifying the effects of the uncertainties in a physical system. In this paper, we propose a stochastic…
The aim of this paper is to develop a numerical scheme to approximate evolving interface problems for parabolic equations based on the abstract evolving finite element framework proposed in (C M Elliott, T Ranner, IMA J Num Anal, 41:3,…
We consider the reliable implementation of an adaptive high-order unfitted finite element method on Cartesian meshes for solving elliptic interface problems with geometrically curved singularities. We extend our previous work on the…
A space discrete approximation to a highly nonlinear reaction-diffusion system endowed with a stochastic dynamical boundary condition is analyzed and the convergence of the discrete scheme to the solution to the corresponding continuum…
Mesh sensitivity of finite element solution for linear elliptic partial differential equations is analyzed. A bound for the change in the finite element solution is obtained in terms of the mesh deformation and its gradient. The bound shows…
Stochastic simulation methods can be applied successfully to model exact spatio-temporally resolved reaction-diffusion systems. However, in many cases, these methods can quickly become extremely computationally intensive with increasing…
In this article, we introduce a finite element method designed for the robust computation of approximate signed distance functions to arbitrary boundaries in two and three dimensions. Our method employs a novel prediction-correction…
This work aims at making a comprehensive contribution in the general area of parametric inference for discretely observed diffusion processes. Established approaches for likelihood-based estimation invoke a time-discretisation scheme for…