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We derive new boundary conditions and implementation procedures for nonlinear initial boundary value problems (IBVPs) with non-zero boundary data that lead to bounded solutions. The new boundary procedure is applied to nonlinear IBVPs in…
The method is proposed for the study of many-point boundary value problems for systems of nonlinear ODE, by reducing them to special equivalent integral equations, and allows us [in contrast with the known method [1]] to consider boundary…
Elliptic partial differential equations arise in many fields of science and engineering such as steady state distribution of heat, fluid dynamics, structural/mechanical engineering, aerospace engineering and seismology etc. In three…
An unsteady three-dimensional boundary element method is developed to provide fast calculations of biological and bio-inspired self-propelled locomotion. The approach uniquely combines an unsteady three-dimensional boundary element method,…
A class of high-order shock-capturing schemes, P$_n$T$_m$-BVD (Deng et al., J. Comp. Phys., 386:323-349, 2019; Comput. & Fluids, 200:104433, 2020.) schemes, have been devised to solve the Euler equations with substantially reduced numerical…
The immersed boundary method is a numerical and mathematical formulation for solving fluid-structure interaction problems. It relies on solving fluid equations on an Eulerian fluid grid and interpolating the resulting velocity back onto…
Computational formulations for large strain, polyconvex, nearly incompressible elasticity have been extensively studied, but research on enhancing solution schemes that offer better tradeoffs between accuracy, robustness, and computational…
A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz…
We present a 3D finite element solver for the nonlinear Poisson-Nernst-Planck (PNP) equations for electrodiffusion, coupled to the Stokes system of fluid dynamics. The model serves as a building block for the simulation of macromolecule…
Nonlinear two-point boundary value problems arise in numerous areas of application. The existence and number of solutions for various cases has been studied from a theoretical standpoint. These results generally rely upon growth conditions…
We propose a component-based (CB) parametric model order reduction (pMOR) formulation for parameterized nonlinear elliptic partial differential equations (PDEs) based on overlapping subdomains. Our approach reads as a constrained…
A moving mesh finite element method is studied for the numerical solution of Bernoulli free boundary problems. The method is based on the pseudo-transient continuation with which a moving boundary problem is constructed and its steady-state…
This work aims to provide standard formulations for direct minimization approaches on various types of static problems of continuum mechanics. Particularly, form-finding problems of tension structures are discussed in the first half and the…
This work presents an alternative view on the numerical simulation of diffusion processes applied to the heat and moisture transfer through porous building materials. Traditionally, by using the finite-difference approach, the…
A shortcoming of existing reachability approaches for nonlinear systems is the poor scalability with the number of continuous state variables. To mitigate this problem we present a simulation-based approach where we first sample a number of…
The periodic standing wave (PSW) method for the binary inspiral of black holes and neutron stars computes exact numerical solutions for periodic standing wave spacetimes and then extracts approximate solutions of the physical problem, with…
Second-order elliptic boundary-value problems defined on curved domains in 2D and 3D arise frequently in practice. A lot of work has gone into developing numerical methods for solving such problems. One of the newest and most promising…
In this paper we derive and analyse efficient and stable numerical methods for accurate numerical simulations of nonlinear dynamic shear ruptures on non-planar faults embedded in 3D elastic solids using dual-paring (DP) summation by parts…
In the paper, we propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for solving Stokes and Navier-Stokes equations. We start with a detailed explanation of the method for the…
A number of important results of studying large deformations of hyper-elastic shells are obtained using discrete methods of mathematical physics. In the present paper, using the variational method for solving nonlinear boundary problems of…