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Introducing flexibility in the time-discretisation mesh can improve convergence and computational time when solving differential equations numerically, particularly when the solutions are discontinuous, as commonly found in control problems…

Optimization and Control · Mathematics 2023-06-27 Lucian Nita , Eduardo M. G. Vila , Marta A. Zagorowska , Eric C. Kerrigan , Yuanbo Nie , Ian McInerney , Paola Falugi

In this work, a time-fractional nonlocal diffusion equation is considered. Based on the $L2$-$1_{\sigma}$ scheme on a graded mesh in time and the standard finite element method (FEM) in space, the fully-discrete $L2$-$1_{\sigma}$ finite…

Numerical Analysis · Mathematics 2023-06-06 Pari J. Kundaliya

In this article, we propose a linearized fully-discrete scheme for solving a time fractional nonlocal diffusion-wave equation of Kirchhoff type. The scheme is established by using the finite element method in space and the $L1$ scheme in…

Numerical Analysis · Mathematics 2023-01-05 Pari J. Kundaliya , Sudhakar Chaudhary

In this paper, we study a fast and linearized finite difference method to solve the nonlinear time-fractional wave equation with multi fractional orders. We first propose a discretization to the multi-term Caputo derivative based on the…

Numerical Analysis · Mathematics 2019-02-22 Pin Lyu , Yuxiang Liang , Zhibo Wang

In this paper, a multiscale approach with partially explicit time discretization is proposed. The idea is to use a partially explicit time scheme, considering a filtration problem in a fractured medium, where the implicit scheme is used for…

Numerical Analysis · Mathematics 2024-02-06 Djulustan Nikiforov

We outline the construction of compatible B-splines on 3D surfaces that satisfy the continuity requirements for electromagnetic scattering analysis with the boundary element method (method of moments). Our approach makes use of Non-Uniform…

Numerical Analysis · Mathematics 2018-04-04 Robert N. Simpson , Zhaowei Liu , Ráfael Vazquez , John A. Evans

This paper deals with a new algorithm called modified trigonometric cubic B-spline differential quadrature method for numerical computation of the time dependent partial differential equations. Specially the numerical computation of the…

Numerical Analysis · Mathematics 2016-11-16 Brajesh Kumar Singh , Pramod Kumar

To extract the approximate solutions in the case of nonlinear fractional order differential equations with the homogeneous and nonhomogeneous boundary conditions, the weighted residual method is embedded here. We exploit three methods such…

Numerical Analysis · Mathematics 2024-04-05 Umme Ruman , Md. Shafiqul Islam

In this paper we address the numerical approximation of linear fourth-order elliptic problems on polygonal meshes. In particular, we present a novel nonconforming virtual element discretization of arbitrary order of accuracy for biharmonic…

Numerical Analysis · Mathematics 2016-11-29 P. F. Antonietti , G. Manzini , M. Verani

We consider an elliptic partial differential equation in non-divergence form with a random diffusion matrix and random forcing term. To address this, we propose a mixed-type continuous finite element discretization in the physical domain,…

Numerical Analysis · Mathematics 2025-12-04 Amireh Mousavi

A linearized numerical scheme is proposed to solve the nonlinear time fractional parabolic problems with time delay. The scheme is based on the standard Galerkin finite element method in the spatial direction, the fractional Crank-Nicolson…

Numerical Analysis · Mathematics 2021-09-10 Lili Li , Mianfu She , Yuanling Niu

We present an efficient B-spline finite element method (FEM) for cloth simulation. While higher-order FEM has long promised higher accuracy, its adoption in cloth simulators has been limited by its larger computational costs while…

Graphics · Computer Science 2026-05-05 Yuqi Meng , Yihao Shi , Kemeng Huang , Zixuan Lu , Ning Guo , Taku Komura , Yin Yang , Minchen Li

The biharmonic equation with Dirichlet and Neumann boundary conditions discretized using the mixed finite element method and piecewise linear (with the possible exception of boundary triangles) finite elements on triangular elements has…

Numerical Analysis · Mathematics 2022-04-21 Oded Stein , Eitan Grinspun , Alec Jacobson , Max Wardetzky

We present weighted quadrature for hierarchical B-splines to address the fast formation of system matrices arising from adaptive isogeometric Galerkin methods with suitably graded hierarchical meshes. By exploiting a local tensor-product…

Numerical Analysis · Mathematics 2022-09-07 Carlotta Giannelli , Tadej Kanduc , Massimiliano Martinelli , Giancarlo Sangalli , Mattia Tani

We propose a non uniform web spline based finite element analysis for elliptic partial differential equation with the gradient type nonlinearity in their principal coefficients like p-laplacian equation and Quasi-Newtonian fluid flow…

Numerical Analysis · Mathematics 2018-07-04 B. V. Rathish Kumar , Ayan Chakraborty

We analyze the convergence of an iterative method for solving the nonlinear system resulting from a natural discretization of the Monge-Amp\`ere equation with $C^1$ conforming approximations. We make the assumption, supported by numerical…

Numerical Analysis · Mathematics 2015-03-17 Gerard Awanou

This paper represents a mixed numerical method for the multi-resolution solution of non-linear partial differential equations based on B-Spline wavelets. The method is based on a second-order finite difference formula combined with the…

Analysis of PDEs · Mathematics 2018-12-27 Seyedhadi Seyedi

This paper discusses lowest-order nonstandard finite element methods for space discretization and explicit and implicit schemes for time discretization of the biharmonic wave equation with clamped boundary conditions. A modified Ritz…

Numerical Analysis · Mathematics 2025-07-16 Neela Nataraj , Ricardo Ruiz-Baier , Aamir Yousuf

We study an approximation method to solve nonlinear multi-term fractional differential equations with initial conditions or boundary conditions. First, we transform the nonlinear multi-term fractional differential equations with initial…

Mathematical Physics · Physics 2013-03-21 Hui-Chol Choe , Yong-Suk Kang

This paper analyzes a space-time finite element method for fractional wave problems. The method uses a Petrov-Galerkin type time-stepping scheme to discretize the time fractional derivative of order $ \gamma $ ($1<\gamma<2$). We establish…

Numerical Analysis · Mathematics 2018-03-12 Binjie Li , Hao Luo , Xiaoping Xie