Decoupling of the quasistatic system of thermoelasticity with Riemann problems on the bounded simply connected domain

Under the displacement and stress satisfying Riemann boundary value condition, the decoupled quasistatic linear thermoelasticity system is discussed on bounded simply connected domain. The quasistatic equilibrium equation is solved by using Riemann boundary value problem theory. Also decoupled temperature equation is studied by applying the contractive mapping principle. Finally, existence and analyticity of the solution are proved.     

Improved T − ψ nodal finite element schemes for eddy current problems

The aim of this paper is to propose improved T − ψ finite element schemes for eddy current problems in the three-dimensional bounded domain with a simply-connected conductor. In order to utilize nodal finite elements in space discretization, we decompose the magnetic field into summation of a vector potential and the gradient of a scalar potential in the conductor; while in the nonconducting domain, we only deal with the gradient of the scalar potential. As distinguished from the traditional coupled scheme with both vector and scalar potentials solved in a discretizing equation system, the proposed decoupled scheme is presented to solve them in two separate equation systems, which avoids solving a saddle-point equation system like the traditional coupled scheme and leads to an important saving in computational effort. The simulation results and the data comparison of TEAM Workshop Benchmark Problem 7 between the coupled and decoupled schemes show the validity and efficiency of the decoupled one.     

Decoupled energy‐law preserving numerical schemes for the Cahn–Hilliard–Darcy system

We study two novel decoupled energy‐law preserving and mass‐conservative numerical schemes for solving the Cahn‐Hilliard‐Darcy system which models two‐phase flow in porous medium or in a Hele–Shaw cell. In the first scheme, the velocity in the Cahn–Hilliard equation is treated explicitly so that the Darcy equation is completely decoupled from the Cahn–Hilliard equation. In the second scheme, an intermediate velocity is used in the Cahn–Hilliard equation which allows for the decoupling. We show that the first scheme preserves a discrete energy law with a time‐step constraint, while the second scheme satisfies an energy law without any constraint and is unconditionally stable. Ample numerical experiments are performed to gauge the efficiency and robustness of our scheme. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 936–954, 2016     

Internal and Boundary Observability Estimates for the Heterogeneous Maxwell's System

Observability estimates for Maxwell's system with variable coefficients are established using the differential geometry method recently developed for scalar wave equations. The main tool is that Maxwell's system is reducible to a perturbed vectorial wave equation with a decoupled principal part.     

Coupled Multi-field and Multi-rate Problems – Numerical Solution and Stability Analysis

The numerical solution of coupled differential equation systems is usually done following a monolithic or a decoupled algorithm. In contrast to the holistic monolithic solvers, the decoupled solution strategies are based on breaking down the system into several subsystems. This results in different characteristics of these families of solvers, e. g., while the monolithic algorithms provide a relatively straight-forward solution framework, unlike their decoupled counterparts, they hinder software re-usability and customisation. This is a drawback for multi-field and multi-rate problems. The reason is that a multi-field problem comprises several subproblems corresponding to interacting subsystems. This suggests exploiting an individual solver for each subproblem. Moreover, for the efficient solution of a multi-rate problem, it makes sense to perform the temporal integration of each subproblem using a time-step size relative to its evolution rate. Nevertheless, decoupled solvers introduce additional errors to the solution and, thus, they must always be accompanied by a thorough stability analysis. Here, tailored solution schemes for the decoupled solution of multi-field and multi-rate problems are proposed. Moreover, the stability behaviour of the solutions obtained from these methods are studied. Numerical examples are solved and the reliability of the outcome of the stability analysis is investigated. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Decoupled characteristic stabilized finite element method for time‐dependent Navier–Stokes/Darcy model

In this article, we propose and analyze a new decoupled characteristic stabilized finite element method for the time‐dependent Navier–Stokes/Darcy model. The key idea lies in combining the characteristic method with the stabilized finite element method to solve the decoupled model by using the lowest‐order conforming finite element space. In this method, the original model is divided into two parts: one is the nonstationary Navier–Stokes equation, and the other one is the Darcy equation. To deal with the difficulty caused by the trilinear term with nonzero boundary condition, we use the characteristic method. Furthermore, as the lowest‐order finite element pair do not satisfy LBB (Ladyzhen‐Skaya‐Brezzi‐Babuska) condition, we adopt the stabilized technique to overcome this flaw. The stability of the numerical method is first proved, and the optimal error estimates are established. Finally, extensive numerical results are provided to justify the theoretical analysis.     

Complete Families of Solutions for the Dirac Equation Using Bicomplex Function Theory and Transmutations

The Dirac equation with a scalar and an electromagnetic potential is considered. In the time-harmonic case and when all the involved functions depend only on two spatial variables it reduces to a pair of decoupled bicomplex Vekua-type equations [8]. Using the technique developed for complex Vekua equations a system of exact solutions for the bicomplex equation is constructed under additional conditions, in particular when the electromagnetic potential is absent and the scalar potential is a function of one Cartesian variable. Introducing a transmutation operator relating the involved bicomplex Vekua equation with the Cauchy-Riemann equation we prove the expansion and the Runge approximation theorems corresponding to the constructed family of solutions.     

求解带有非线性边界条件的涡流方程的A-φ解耦有限元格式

本文研究边界条件符合幂指数型非线性关系H × n = n × (|E × n|^α-1E × n)(0 < α ≤ 1)的涡流方程.使用A-φ耦合有限元格式数值求解这类问题具有较高精度,但计算开销大. A-φ解耦有限元计算格式能够在每个时间步上分别求解矢量A和标量φ,以此降低计算规模,提高计算效率.我们证明了解耦格式中解的存在唯一性,并且给出了它的误差估计.最后给出的数值实验证明了本文所提供的解耦算法是稳定和有效的.     

Superlinearly convergent PCG algorithms for some nonsymmetric elliptic systems

A preconditioned conjugate gradient method is applied to finite element discretizations of some nonsymmetric elliptic systems. Mesh independent superlinear convergence is proved, which is an extension of a similar earlier result from a single equation to systems. The proposed preconditioning method involves decoupled preconditioners, which yields small and parallelizable auxiliary problems.     

Boundary integral equation method in thermoelasticity: part II crack analysis

The boundary integral equation formulation of thermoelasticity problems from part I is applied to crack problems in both finite and infinite thermoelastic bodies. For a flat crack in an infinite body the normal and tangential crack opening displacement are decoupled. Transient and steady state problems of thermoelasticity, as well as stationary problems, are considered.     

Decoupled modified characteristics finite element method for the time dependent Navier–Stokes/Darcy problem

In this paper, a modified characteristics finite element method for the time dependent Navier–Stokes/Darcy problem with the Beavers–Joseph–Saffman interface condition is presented. In this method, the Navier–Stokes/Darcy equation is decoupled into two equations, one is the Navier–Stokes equation, the other is the Darcy equation, and the Navier–Stokes equation is solved by the modified characteristics finite element method. The theory analysis shows that this method has a good convergence property. In order to show the effect of our method, some numerical results was presented. The numerical results show that this method is highly efficient. Copyright © 2013 John Wiley & Sons, Ltd.     

Efficient and Energy Stable Scheme for an Anisotropic Phase-Field Dendritic Crystal Growth Model Using the Scalar Auxiliary Variable (SAV) Approach

The phase-field dendritic crystal growth model is a highly nonlinear system that couples the anisotropic Allen-Cahn type equation and the heat equation. By combining the recently developed SAV (Scalar Auxiliary Variable) method with the linear stabilization approach, as well as a special decoupling technique, we arrive at a totally decoupled, linear, and unconditionally energy stable scheme for solving the dendritic model. We prove its unconditional energy stability rigorously and present various numerical simulations to demonstrate the stability and accuracy.     

A (T,ψ)‐ψe decoupled scheme for a time‐dependent multiply‐connected eddy current problem

The aim of this paper is to develop a fully discrete ( T ,ψ)‐ψ_e finite element decoupled scheme to solve time‐dependent eddy current problems with multiply‐connected conductors. By making ‘cuts’ and setting jumps of ψ_e across the cuts in nonconductive domain, the uniqueness of ψ_e is guaranteed. Distinguished from the traditional T ‐ ψ method, our decoupled scheme solves the potentials T and ψ‐ ψ_e separately in two different simple equation systems, which avoids solving a saddle‐point equation system and leads to a remarkable reduction in computational efforts. The energy‐norm error estimate of the fully discrete decoupled scheme is provided. Finally, the scheme is applied to solve two benchmark problems—TEAM Workshop Problems 7 and IEEJ model. Copyright © 2013 John Wiley & Sons, Ltd.     

Stability Analysis of Decoupled Solution Strategies for Coupled Multi-field Problems – A General Framework

A coupled partial differential equation (PDE) system, stemming from the mathematical modelling of a coupled phenomenon, is usually solved numerically following a monolithic or a decoupled solution method. In spite of the potential unconditional stability offered by monolithic solvers, their usage for solving complex problems sometimes proves cumbersome. This has motivated the development of various partitioned and staggered solution strategies, generally known as decoupled solution schemes. To this end, the problem is broken down into several isolated yet communicating sub-problems that are independently advanced in time, possibly by different integrators. Nevertheless, using a decoupled solver introduces additional errors to the system and, therefore, may jeopardise the stability of the solution [1]. Consequently, to scrutinise the stability of the solution scheme becomes a pertinent step in proposing decoupled solution strategies. Here, we endeavour to present a practical stability analysis algorithm, which can readily be used to reveal the stability condition of numerical solvers. To illustrate its capabilities, the algorithm is then utilised for the stability analysis of solution schemes applied to multi variate coupled PDE systems resulting from the mathematical modelling of surface- and volume-coupled multi-field problems. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Decoupled Mixed Element Methods for Fourth Order Elliptic Optimal Control Problems with Control Constraints

In this paper, we study the finite element methods for distributed optimal control problems governed by the biharmonic operator. Motivated from reducing the regularity of solution space, we use the decoupled mixed element method which was used to approximate the solution of biharmonic equation to solve the fourth order optimal control problems. Two finite element schemes, i.e., Lagrange conforming element combined with full control discretization and the nonconforming Crouzeix-Raviart element combined with variational control discretization, are used to discretize the decoupled optimal control system. The corresponding a priori error estimates are derived under appropriate norms which are then verified by extensive numerical experiments.     

Decoupling of the N-soliton solution for a new discrete MKdV equation

We prove that the N-soliton solution for a new discrete MKdV equation can be decoupled into N components. These components satisfy the interacting new discrete MKdV equation, and approach single soliton solutions as t→∞. We present concrete expressions of soliton components in the case of N = 2, and illustrate the results.     

Travelling Wave Solutions and Conservation Laws of the (2+1)-dimensional Broer-Kaup-Kupershmidt Equatio

The travelling wave solutions and conservation laws of the (2+1)-dimensional Broer-Kaup-Kupershmidt (BKK) equation are considered in this paper. Under the travelling wave frame, the BKK equation is transformed to a system of ordinary differential equations (ODEs) with two dependent variables. Therefore, it happens that one dependent variable $u$ can be decoupled into a second order ODE that corresponds to a Hamiltonian planar dynamical system involving three arbitrary constants. By using the bifurcation analysis, we obtain the bounded travelling wave solutions $u$, which include the kink, anti-kink and periodic wave solutions. Finally, the conservation laws of the BBK equation are derived by employing the multiplier approach.     

Numerical approximation of the Oldroyd‐B model by the weighted least‐squares/discontinuous Galerkin method

We consider a numerical method for the Oldroyd‐B model of viscoelastic fluid flows by a combination of the weighted least‐squares (WLS) method and the discontinuous Galerkin (DG) finite element method. The constitutive equation is decoupled from the momentum and continuity equations, and the approximate solution is computed iteratively by solving the Stokes problem and a linearized constitutive equation using WLS and DG, respectively. An a priori error estimate for the WLS/DG method is derived and numerical results supporting the estimate are presented. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013     

A conservative SPH scheme using exact projection with semi-analytical boundary method for free-surface flows

This paper presents a novel SPH scheme for modelling incompressible and divergence-free flow with a free surface (IDFSPH) associated with semi-analytical wall boundary conditions. In line with the projection method, the velocity field is decoupled from the pressure field in the momentum equation. A Poisson equation, serving as the pressure solver, is obtained by which pressure field is decoupled completely from the velocity field. In particular, an exact projection scheme is deployed to fulfil the requirement of the divergence-free velocity field. The condition of incompressibility is satisfied by iteratively updating the density field till the convergence. The two-equation k–ε model is employed to describe the turbulence effects in Newtonian flows. It is shown that the discretised SPH schemes have the feature of both linear and angular momentum conservations. The semi-analytical wall method implements the appropriate integrals to evaluate the boundary contributions to the mass and momentum equations. In comparison to the boundary particle methods, it can greatly enhance the feasibility and efficiency with the complex geometries. The algorithm presented within this paper is applied to several academic test cases for which either analytical results or simulations with other methods are available. The comparisons verify that this scheme is provided with convincing efficiency and extensive applicability.