Related papers: Neural network-enhanced $hr$-adaptive finite eleme…
This paper proposes an efficient algorithm for solving the Hartree--Fock equation combining a multilevel correction scheme with an adaptive refinement technique to improve computational efficiency. The algorithm integrates a multilevel…
In this paper, we study adaptive finite element approximations in a perturbation framework, which makes use of the existing adaptive finite element analysis of a linear symmetric elliptic problem. We prove the convergence and complexity of…
Machine learning is evolving towards high-order models that necessitate pre-training on extensive datasets, a process associated with significant overheads. Traditional models, despite having pre-trained weights, are becoming obsolete due…
We present a novel, and effective, approach to achieve optimal mesh relocation in finite element methods (FEMs). The cost and accuracy of FEMs is critically dependent on the choice of mesh points. Mesh relocation (r-adaptivity) seeks to…
We propose an efficient method for the numerical approximation of a general class of two dimensional semilinear parabolic problems on polygonal meshes. The proposed approach takes advantage of the properties of the serendipity version of…
The conformal formulation of the Einstein constraint equations is first reviewed, and we then consider the design, analysis, and implementation of adaptive multilevel finite element-type numerical methods for the resulting coupled nonlinear…
We propose an adaptive iteratively linearized finite element method (AILFEM) in the context of strongly monotone nonlinear operators in Hilbert spaces. The approach combines adaptive mesh-refinement with an energy-contractive linearization…
We present an algorithm for $hp$-adaptive collocation-based mesh-free numerical analysis of partial differential equations. Our solution procedure follows a well-established iterative solve-estimate-mark-refine paradigm. The solve phase…
For a singularly perturbed elliptic model problem with two small parameters, we analyze finite element methods of any order on a Bakhvalov-type mesh. For convergence analysis, we construct a new interpolation by using the characteristics of…
We analyze optimal complexity of adaptive finite element methods (AFEMs) for general second-order linear elliptic partial differential equations (PDEs) in the Lax-Milgram setting. To this end, we formulate an adaptive algorithm which steers…
We propose a new practical adaptive refinement strategy for $hp$-finite element approximations of elliptic problems. Following recent theoretical developments in polynomial-degree-robust a posteriori error analysis, we solve two types of…
A neural network architecture is presented that exploits the multilevel properties of high-dimensional parameter-dependent partial differential equations, enabling an efficient approximation of parameter-to-solution maps, rivaling…
Localized features such as singularities, sharp gradients, discontinuities, and moving sources require adaptive finite element discretizations. Conventional refinement strategies introduce significant computational overhead through…
Mesh-based simulations provide high-fidelity solutions to partial differential equations (PDEs), but achieving such accuracy typically requires fine meshes, leading to substantial computational overhead. Super-resolution techniques aim to…
In this article, we design and analyze a hybrid high-order (HHO) finite element approximation for the solution of a nonlocal nonlinear problem of Kirchhoff type. The HHO method involves arbitrary-order polynomial approximations on…
In this paper we address three aspects of nonlinear computational homogenization of elastic solids by two-scale finite element methods. First, we present a nonlinear formulation of the finite element heterogeneous multiscale method FE-HMM…
In this work we develop and analyze an adaptive finite element method for efficiently solving electrical impedance tomography -- a severely ill-posed nonlinear inverse problem for recovering the conductivity from boundary voltage…
We develop an interpolation-based modeling framework for parameter-dependent partial differential equations arising in control, inverse problems, and uncertainty quantification. The solution is discretized in the physical domain using…
Modeling of physical systems includes extensive use of software packages that implement the accurate finite element method for solving differential equations considered along with the appropriate initial and boundary conditions. When the…
In this paper, a novel adaptive finite element method is proposed to solve the Kohn-Sham equation based on the moving mesh (nonnested mesh) adaptive technique and the augmented subspace method. Different from the classical self-consistent…