Related papers: Iterative solvers in adaptive FEM
We consider a linear elliptic partial differential equation (PDE) with a generic uniformly bounded parametric coefficient. The solution to this PDE problem is approximated in the framework of stochastic Galerkin finite element methods. We…
In this work, we formulate and analyze a geometric multigrid method for the iterative solution of the discrete systems arising from the finite element discretization of symmetric second-order linear elliptic diffusion problems. We show that…
This article presents new immersed finite element (IFE) methods for solving the popular second order elliptic interface problems on structured Cartesian meshes even if the involved interfaces have nontrivial geometries. These IFE methods…
Elliptic Partial Differential Equations (PDEs) play a central role in computing the equilibrium conditions of physical problems (heat, gravitation, electrostatics, etc.). Efficient solutions to elliptic PDEs are also relevant to computer…
The realization of a standard Adaptive Finite Element Method (AFEM) preserves the mesh conformity by performing a completion step in the refinement loop: in addition to elements marked for refinement due to their contribution to the global…
In this work, we bridge standard adaptive mesh refinement and coarsening on scalable octree background meshes and robust unfitted finite element formulations for the automatic and efficient solution of large-scale nonlinear solid mechanics…
Algorithms that promise to leverage resources of quantum computers efficiently to accelerate the finite element method have emerged. However, the finite element method is usually incorporated into a high-level numerical scheme which allows…
This article is concerned with the numerical solution of convex variational problems. More precisely, we develop an iterative minimisation technique which allows for the successive enrichment of an underlying discrete approximation space in…
Partial Differential Equations (PDEs) are fundamental tools for modeling physical phenomena, yet most PDEs of practical interest cannot be solved analytically and require numerical approximations. The feasibility of such numerical methods,…
The finite element method (FEM) is a well-established numerical method for solving partial differential equations (PDEs). However, its mesh-based nature gives rise to substantial computational costs, especially for complex multiscale…
The finite element methods (FEM) are important techniques in engineering for solving partial differential equations, but they depend heavily on element shape quality for stability and good performance. In this paper, we introduce the…
This study presents an adaptive coupling peridynamic least-square minimization with the finite element method (PDLSM-FEM) for fracture analysis. The presented method utilizes the PDLSM modeling discontinuities while maximizing the FEM…
The solution for non-linear, complex partial differential Equations (PDEs) is achieved through numerical approximations, which yield a linear system of equations. This approach is prevalent in Computational Fluid Dynamics (CFD), but it…
In this work, we develop an adaptive nonconforming finite element algorithm for the numerical approximation of phase-field parameterized topology optimization governed by the Stokes system. We employ the conforming linear finite element…
Multiscale and multiphysics problems need novel numerical methods in order for them to be solved correctly and predictively. To that end, we develop a wavelet based technique to solve a coupled system of nonlinear partial differential…
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…
We develop and evaluate a method for learning solution operators to nonlinear problems governed by partial differential equations (PDEs). The approach is based on a finite element discretization and aims at representing the solution…
The efficient approximation of quantity of interest derived from PDEs with lognormal diffusivity is a central challenge in uncertainty quantification. In this study, we propose a multilevel quasi-Monte Carlo framework to approximate…
Neural networks have shown significant potential in solving partial differential equations (PDEs). While deep networks are capable of approximating complex functions, direct one-shot training often faces limitations in both accuracy and…
In this paper, we discuss adaptive approximations of an elliptic eigenvalue optimization problem in a phase-field setting by a conforming finite element method. An adaptive algorithm is proposed and implemented in several two dimensional…