Related papers: Using p-Refinement to Increase Boundary Derivative…
We introduce a novel method for bounding high-order multi-dimensional polynomials in finite element approximations. The method involves precomputing optimal piecewise-linear bounding boxes for polynomial basis functions, which can then be…
The objective of this study is to address the difficulty of simplifying the geometric model in which a differential problem is formulated, also called defeaturing, while simultaneously ensuring that the accuracy of the solution is…
We study the discretization of an elliptic partial differential equation, posed on a two- or three-dimensional domain with smooth boundary, endowed with a generalized Robin boundary condition which involves the Laplace-Beltrami operator on…
In this article we develop a high order accurate method to solve the incompressible boundary layer equations in a provably stable manner.~We first derive continuous energy estimates,~and then proceed to the discrete setting.~We formulate…
Upper bound limit analysis allows one to evaluate directly the ultimate load of structures without performing a cumbersome incremental analysis. In order to numerically apply this method to thin plates in bending, several authors have…
The finite cell method is a highly flexible discretization technique for numerical analysis on domains with complex geometries. By using a non-boundary conforming computational domain that can be easily meshed, automatized computations on a…
We present a cut finite element method for shape optimization in the case of linear elasticity. The elastic domain is defined by a level-set function, and the evolution of the domain is obtained by moving the level-set along a velocity…
We propose a novel a posteriori error estimator for conforming finite element discretizations of two- and three-dimensional Helmholtz problems. The estimator is based on an equilibrated flux that is computed by solving patchwise mixed…
For shape optimization problems, governed by elliptic equations with Dirichlet boundary condition and random coefficients, we utilize a penalization technique to get the approximate problem. We consider that uncertainties exists in the…
A regularization algorithm allowing random noise in derivatives and inexact function values is proposed for computing approximate local critical points of any order for smooth unconstrained optimization problems. For an objective function…
We derive a priori error estimates for Nitsche's method applied to elliptic problems on approximate domains. Such approximations arise, for example, in unfitted finite element methods, data-driven simulations, and evolving domain problems,…
We consider the finite element discretization and the iterative solution of singularly perturbed elliptic reaction-diffusion equations in three-dimensional computational domains. These equations arise from the optimality conditions for…
We show that an optimal finite element mesh refinement algorithm for a prototypical elliptic PDE can be learned by a recurrent neural network with a fixed number of trainable parameters independent of the desired accuracy and the input…
The boundary control problem is a non-convex optimization and control problem in many scientific domains, including fluid mechanics, structural engineering, and heat transfer optimization. The aim is to find the optimal values for the…
We use uniform $W^{2,p}$ estimates to obtain corrector results for periodic homogenization problems of the form $A(x/\varepsilon):D^2 u_{\varepsilon} = f$ subject to a homogeneous Dirichlet boundary condition. We propose and rigorously…
We extend a localized model order reduction method for the distributed finite element solution of elliptic boundary value problems in the cloud. We give a computationally efficient technique to compute the required inner product matrices…
Approximating solutions of ordinary and partial differential equations constitutes a significant challenge. Based on functional expressions that inherently depend on neural networks, neural forms are specifically designed to precisely…
We consider the problem of numerically approximating the solutions to an elliptic partial differential equation (PDE) for which the boundary conditions are lacking. To alleviate this missing information, we assume to be given measurement…
We analyze the convergence of degenerate approximations to Green's function of elliptic boundary value problems with high-contrast coefficients. It is shown that the convergence is independent of the contrast if the error is measured with…
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…