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A semidiscrete Galerkin finite element method applied to time-fractional diffusion equations with time-space dependent diffusivity on bounded convex spatial domains will be studied. The main focus is on achieving optimal error results with…
The recently proposed soft finite element method (SoftFEM) reduces the stiffness (condition numbers), consequently improving the overall approximation accuracy. The method subtracts a least-square term that penalizes the gradient jumps…
In this work we investigate the numerical identification of the diffusion coefficient in elliptic and parabolic problems using neural networks. The numerical scheme is based on the standard output least-squares formulation where the…
We introduce a novel hybrid methodology combining classical finite element methods (FEM) with neural networks to create a well-performing and generalizable surrogate model for forward and inverse problems. The residual from finite element…
A previously developed quantum reduced-order model is revised and applied, together with the domain decomposition, to develop the quantum element method (QEM), a methodology for fast and accurate simulation of quantum eigenvalue problems.…
A boundary element method (BEM) simulation is used to compare the efficiency of numerical inverse Laplace transform strategies, considering general requirements of Laplace-space numerical approaches. The two-dimensional BEM solution is used…
Stability and optimal convergence analysis of a non-uniform implicit-explicit L1 finite element method (IMEX-L1-FEM) is studied for a class of time-fractional linear partial differential/integro-differential equations with non-self-adjoint…
A novel boundary element method (BEM) removes the classical dependence on explicit fundamental solutions and extends quasi-optimal BEM discretisations to strongly elliptic operators with variable coefficients. The approach constructs a…
This paper is concerned with error estimates of the fully discrete generalized finite element method (GFEM) with optimal local approximation spaces for solving elliptic problems with heterogeneous coefficients. The local approximation…
Neural operators (NOs) struggle with high-contrast multiscale partial differential equations (PDEs), where fine-scale heterogeneities cause large errors. To address this, we use the Generalized Multiscale Finite Element Method (GMsFEM) that…
The adaptive nonconforming Morley finite element method (FEM) approximates a regular solution to the von K\'{a}rm\'{a}n equations with optimal convergence rates for sufficiently fine triangulations and small bulk parameter in the D\"orfler…
In this paper, the generalized finite element method (GFEM) for solving second order elliptic equations with rough coefficients is studied. New optimal local approximation spaces for GFEMs based on local eigenvalue problems involving a…
Topological optimization finds a material density distribution minimizing a functional of the solution of a partial differential equation (PDE), subject to a set of constraints (typically, a bound on the volume or mass of the material).…
Motivated by problems where the response is needed at select localized regions in a large computational domain, we devise a novel finite element discretization that results in exponential convergence at pre-selected points. The two key…
The Finite Element Method (FEM) is a well-established procedure for computing approximate solutions to deterministic engineering problems described by partial differential equations. FEM produces discrete approximations of the solution with…
We propose an efficient algorithm for the approximation of fractional integrals by using Runge--Kutta based convolution quadrature. The algorithm is based on a novel integral representation of the convolution weights and a special…
In this paper, we propose a way to solve partial differential equations (PDEs) by combining machine learning techniques and the finite element method called Phi-FEM. For that, we use the Fourier Neural Operator (FNO), a learning mapping…
The $hp$-adaptive finite element method (FEM) - where one independently chooses the mesh size ($h$) and polynomial degree ($p$) to be used on each cell - has long been known to have better theoretical convergence properties than either $h$-…
This paper deals with the asymptotic behavior and FEM error analysis of a class of strongly damped wave equations using a semidiscrete finite element method in spatial directions combined with a finite difference scheme in the time…
In this paper, based on the combination of finite element mesh and neural network, a novel type of neural network element space and corresponding machine learning method are designed for solving partial differential equations. The…