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We consider a finite element method for elliptic equation with heterogeneous and possibly high-contrast coefficients based on primal hybrid formulation. A space decomposition as in FETI and BDCC allows a sequential computations of the…
These notes offer a unified introduction to spectral methods for the study of complex systems. They are intended as an operative manual rather than a theorem-proof textbook: the emphasis is on tools, identities, and perspectives that can be…
The modeling of electric machines and power transformers typically involves systems of nonlinear magnetostatics or -quasistatics, and their efficient and accurate simulation is required for the reliable design, control, and optimization of…
This paper introduces an extension of the Morley element for approximating solutions to biharmonic equations. Traditionally limited to piecewise quadratic polynomials on triangular elements, the extension leverages weak Galerkin finite…
The main purpose of this article is to facilitate the implementation of space-time finite element methods in four-dimensional space. In order to develop a finite element method in this setting, it is necessary to create a numerical…
We introduce new numerical integration operators which compose the mass and stiffness matrices of a modified spectral element method for simulation of elastic wave propagation. While these operators use the same quadrature nodes as does the…
Hypergraphs offer flexible and robust data representations for many applications, but methods that work directly on hypergraphs are not readily available and tend to be prohibitively expensive. Much of the current analysis of hypergraphs…
We describe a high order technique to generate quadrilateral decompositions and meshes for complex two dimensional domains using spectral elements in a field guided procedure. Inspired by cross field methods, we never actually compute…
The concept of effective order is a popular methodology in the deterministic literature for the construction of efficient and accurate integrators for differential equations over long times. The idea is to enhance the accuracy of a…
Higher-order spectra (Brillinger's polyspectra) offer powerful methods for solving critical problems in signal processing and data analysis. Despite their significant potential, their practical use has remained limited due to unresolved…
This study proposes a high-order multi-scale method tailored for time-dependent nonlinear thermo-electro-mechanical coupling problems of composite structures with highly spatial heterogeneity, which incorporate temperature-dependent…
We present higher-order piecewise continuous finite element methods for solving a class of interface problems in two dimensions. The method is based on correction terms added to the right-hand side in the standard variational formulation of…
Post-processing techniques are essential tools for enhancing the accuracy of finite element approximations and achieving superconvergence. Among these, recovery techniques stand out as vital methods, playing significant roles in both…
We consider a fast, data-sparse directional method to realize matrix-vector products related to point evaluations of the Helmholtz kernel. The method is based on a hierarchical partitioning of the point sets and the matrix. The considered…
Higher-order accurate solution to electromagnetic scattering problems are obtained at reduced computational cost in a {\it p}-variable finite volume time domain method. Spatial operators of lower, including first-order accuracy, are…
Equations arising in General Relativity are usually too complicated to be solved analytically and one has to rely on numerical methods to solve sets of coupled partial differential equations. Among the possible choices, this paper focuses…
We prove convergence of the spectral element method for piecewise polynomial collocation applied to periodic boundary value problems for functional differential equations. In particular, we prove that the numerical collocation solution…
The purpose of this work is to introduce and analyze a numerical scheme to efficiently solve boundary value problems involving the spectral fractional Laplacian. The approach is based on a reformulation of the problem posed on a…
The matrix element method utilizes ab initio calculations of probability densities as powerful discriminants for processes of interest in experimental particle physics. The method has already been used successfully at previous and current…
The need for optically multi-functional micro- and nano-structures is growing in various fields. Designing such structures is impeded by the lack of computationally low-cost algorithms. In this study, we present a hybrid design scheme,…