A Kinetic Energy Preserving DG Scheme Based on Gauss–Legendre Points

In the context of numerical methods for conservation laws, not only the preservation of the primary conserved quantities can be of interest, but also the balance of secondary ones such kinetic energy in case of the Euler equations of gas dynamics. In this work, we construct a kinetic energy preserving discontinuous Galerkin method on Gauss–Legendre nodes based on the framework of summation-by-parts operators. For a Gauss–Legendre point distribution, boundary terms require special attention. In fact, stability problems will be demonstrated for a combination of skew-symmetric and boundary terms that disagrees with exclusively interior nodal sets. We will theoretically investigate the required form of the corresponding boundary correction terms in the skew-symmetric formulation leading to a conservative and consistent scheme. In numerical experiments, we study the order of convergence for smooth solutions, the kinetic energy balance and the behaviour of different variants of the scheme applied to an acoustic pressure wave and a viscous shock tube. Using Gauss–Legendre nodes results in a more accurate approximation in our numerical experiments for viscous compressible flow. Moreover, for two-dimensional decaying homogeneous turbulence, kinetic energy preservation yields a better representation of the energy spectrum.     

A mixed–primal finite element method for the Boussinesq problem with temperature-dependent viscosity

In this paper we focus on the analysis of a mixed finite element method for a class of natural convection problems in two dimensions. More precisely, we consider a system based on the coupling of the steady-state equations of momentum (Navier–Stokes) and thermal energy by means of the Boussinesq approximation (coined the Boussinesq problem), where we also take into account a temperature dependence of the viscosity of the fluid. The construction of this finite element method begins with the introduction of the pseudostress and vorticity tensors, and a mixed formulation for the momentum equations, which is augmented with Galerkin-type terms, in order to deal with the non-linearity of these equations and the convective term in the energy equation, where a primal formulation is considered. The prescribed temperature on the boundary becomes an essential condition, which is weakly imposed, leading us to the definition of the normal heat flux through the boundary as a Lagrange multiplier. We show that this highly coupled problem can be uncoupled and analysed as a fixed-point problem, where Banach and Brouwer theorems will help us to provide sufficient conditions to ensure well-posedness of the problems arising from the continuous and discrete formulations, along with several applications of continuous injections guaranteed by the Rellich–Kondrachov theorem. Finally, we show some numerical results to illustrate the performance of this finite element method, as well as to prove the associated rates of convergence.     

Well-posedness in the generalized sense for boundary layer suppressing boundary conditions

The time-dependent Navier-Stokes equation for incompressible fluid flow together with new boundary layer suppressing boundary conditions for open boundaries is investigated. In these new boundary conditions one typically prescribes a high-order derivative of some of the dependent variables. We prove that these boundary conditions give rise to a problem that is well posed in the generalized sense. This means that there exists a unique smooth solution of the linearized problem and that this solution can be estimated by data.     

Numerical investigation based on radial basis function–finite-difference (RBF–FD) method for solving the Stokes–Darcy equations

A meshless method is presented to numerically study an interface problem between a flow in a porous medium governed by Darcy equations and a fluid flow, governed by Stokes equations. In fact, the domain of the problem has two parts, one governed by Stokes equations and the another governed by Darcy law. Governing equations on these two parts are mutually coupled by interface conditions. The approximation solution is based on local radial basis function–finite-difference (RBF–FD) which is carried out within a small influence domain instead of a global one. By this strategy, the final linear system is more sparse and well posed than the global one. Several numerical results are provided to illustrate the good performance of the proposed scheme.

     

L 2 estimates for Chebyshev collocation

We study Chebyshev collocation when applied to a system of symmetric hyperbolic equations on a finite domain with general boundary conditions. We show that the use of orthogonal projections in theL ₂ norm in order to smooth out the higher modes and to implement boundary conditions leads to a stable numerical approximation in theL ₂ norm; the stability estimate corresponds to the estimate of the continuous problem. For constant coefficient systems the method reduces to an efficient implementation of Legendre-Galerkin.     

Element free method for static and free vibration analysis of spatial thin shell structures

The implementation of the element free Galerkin method (EFG) for spatial thin shell structures is presented in this paper. Both static deformation and free vibration analyses are considered. The formulation of the discrete system equations starts from the governing equations of stress resultant geometrically exact theory of shear flexible shells. Moving least squares approximation is used in both the construction of shape functions based on arbitrarily distributed nodes as well as in the surface approximation of general spatial shell geometry. Discrete system equations are obtained by incorporating these interpolations into the Galerkin weak form. The formulation is verified through numerical examples of static stress analysis and frequency analysis of spatial thin shell structures. For static load analysis, essential boundary conditions are enforced through penalty method and Lagrange multipliers while boundary conditions for frequency analysis are imposed through a weak form using orthogonal transformation techniques. The EFG results compare favorably with closed-form solutions and that of finite element analyses.     

High-Order Numerical Methods for 2D Parabolic Problems in Single and Composite Domains

In this work, we discuss and compare three methods for the numerical approximation of constant- and variable-coefficient diffusion equations in both single and composite domains with possible discontinuity in the solution/flux at interfaces, considering (i) the Cut Finite Element Method; (ii) the Difference Potentials Method; and (iii) the summation-by-parts Finite Difference Method. First we give a brief introduction for each of the three methods. Next, we propose benchmark problems, and consider numerical tests—with respect to accuracy and convergence—for linear parabolic problems on a single domain, and continue with similar tests for linear parabolic problems on a composite domain (with the interface defined either explicitly or implicitly). Lastly, a comparative discussion of the methods and numerical results will be given.     

A Mixed and Nonconforming FEM with Nonmatching Meshes for a Coupled Stokes-Darcy Model

In this paper, we study numerical methods for a coupled Stokes-Darcy model. This model is composed by Stokes equations in the fluid domain and Darcy??s law in the porous media domain, coupling together through certain interface conditions. By introducing a Lagrange multiplier, the coupled model is formed into a saddle point problem. A nonconforming mixed finite element method with nonmatching meshes is proposed to solve this coupled problem. The well-posedness of the discrete problem is proved. Moreover, we derive the a priori error estimates of the proposed finite element method. Numerical examples are also given to confirm the theoretical results.     

A minimum principle for smooth first-order distributed systems

The problem of characterizing optimal controls for a class of distributed parameter systems is considered. The system dynamics are characterized mathematically by a finite number of coupled partial differential equations involving first-order time and space derivatives of the state variables. Boundary conditions on the state are in the form of a finite number of algebraic relations between the state and boundary control variables. Multiple distributed controls extending over the entire spatial region occupied by the system are also included. The performance index is an integral over the spatial domain of penalty functions on the terminal state and on the distributed state and controls. Under certain differentiability and well-posedness assumptions, variational methods are used to derive first- and second-order necessary conditions for a control which minimizes the performance index. Of particular interest are conditions on the boundary value of the costate and on the optimal boundary controls.     

Modelling and simulation of a catalytic membrane reactor

A catalytic membrane reactor modelling is established through mass balance equations in different control volumes. Two different strategies were followed to solve the parabolic differential system representing the mass balance. The orthogonal collocation method in two spatial dimensions is first used to solve the elliptic form of the equation corresponding to steady state. Secondly, a structural approximation method, i.e. finite difference method is used to solve the global dynamic system. The approximation problem is studied by using some convergence results of discretising approximation of a linear continuous operator in a Banach space. The two approximation methods give sound modelling of this new type of reactor for numerical solutions study and approximative structure study.     

Finite element analysis of nonlocal coupled parabolic problem using Newton’s method

In this article, we propose finite element method to approximate the solution of a coupled nonlocal parabolic system. An important issue in the numerical solution of nonlocal problems while using the Newton’s method is related to its structure. Indeed, unlike the local case the Jacobian matrix is sparse and banded, the nonlocal term makes the Jacobian matrix dense. As a consequence computations consume more time and space in contrast to local problems. To overcome this difficulty we reformulate the discrete problem and then apply the Newton’s method. We discuss the well-posedness of the weak formulation at continuous as well as at discrete levels. We derive a priori error estimates for both semi-discrete and fully-discrete formulations. Results based on usual finite element method are provided to confirm the theoretical estimates.     

Nonlinear control of a coupled PDE/ODE system modeling a switched power converter with a transmission line

We consider an infinite dimensional system modeling a boost converter connected to a load via a transmission line. The governing equations form a system coupling the telegraph partial differential equation with the ordinary differential equations modeling the converter. The coupling is given by the boundary conditions and the nonlinear controller we introduce. We design a nonlinear saturating control law using a Lyapunov function for the averaged model of the system. The main results give the well-posedness and stability properties of the obtained closed loop system.     

Mumford-Shah on the Move: Region-Based Segmentation on Deforming Manifolds with Application to 3-D Reconstruction of Shape and Appearance from Multi-View Images

We address the problem of estimating the shape and appearance of a scene made of smooth Lambertian surfaces with piecewise smooth albedo. We allow the scene to have self-occlusions and multiple connected components. This class of surfaces is often used as an approximation of scenes populated by man-made objects. We assume we are given a number of images taken from different vantage points. Mathematically this problem can be posed as an extension of Mumford and Shah’s approach to static image segmentation to the segmentation of a function defined on a deforming surface. We propose an iterative procedure to minimize a global cost functional that combines geometric priors on both the shape of the scene and the boundary between smooth albedo regions. We carry out the numerical implementation in the level set framework.     

Interaction of Waves with Frictional Interfaces Using Summation-by-Parts Difference Operators: Weak Enforcement of Nonlinear Boundary Conditions

We present a high-order difference method for problems in elastodynamics involving the interaction of waves with highly nonlinear frictional interfaces. We restrict our attention to two-dimensional antiplane problems involving deformation in only one direction. Jump conditions that relate tractions on the interface, or fault, to the relative sliding velocity across it are of a form closely related to those used in earthquake rupture models and other frictional sliding problems. By using summation-by-parts (SBP) finite difference operators and weak enforcement of boundary and interface conditions, a strictly stable method is developed. Furthermore, it is shown that unless the nonlinear interface conditions are formulated in terms of characteristic variables, as opposed to the physical variables in terms of which they are more naturally stated, the semi-discretized system of equations can become extremely stiff, preventing efficient solution using explicit time integrators.     

An asymptotic numerical method for fourth order singular perturbation problems with a discontinuous source term

Singularly perturbed two-point boundary-value problems (BVPs) for fourth-order ordinary differential equations (ODEs) with a small positive parameter multiplying the highest derivative with a discontinuous source term is considered. The given fourth-order BVP is transformed into a system of weakly coupled systems of two second-order ODEs, one without the parameter and the other with the parameter ? multiplying the highest derivative, and suitable boundary conditions. In this paper a computational method for solving this system is presented. In this method we first find the zero-order asymptotic approximation expansion of the solution of the weakly coupled system. Then the system is decoupled by replacing the first component of the solution by its zero-order asymptotic approximation expansion of the solution in the second equation. Then the second equation is solved by the numerical method, which is constructed for this problem and which involves an appropriate piecewise-uniform mesh.     

Coupling of continuous and discontinuous Galerkin methods for transport problems

In this paper, we formulate a coupled discontinuous/continuous Galerkin method for the numerical solution of convection–diffusion (transport) equations, where convection may be dominant. One motivation for this approach is to use a discontinuous method where the solution is rough, e.g., in regions of high gradients, and use a continuous method where the solution is smooth. In this approach, the domain is decomposed into two regions, and appropriate transmission conditions are applied at the interface between regions. In one region, a local discontinuous Galerkin method is applied, and in the other region a standard continuous Galerkin method is employed. Stability and a priori error estimates for the coupled method are derived, and numerical results in one space dimension are given for smooth problems and problems with sharp fronts.     

Trefftz approximations in complex media: Accuracy and applications

Approximations by Trefftz functions are rapidly gaining popularity in the numerical solution of boundary value problems of mathematical physics. By definition, these functions satisfy locally, in weak form, the underlying differential equations of the problem, which often results in high-order or even exponential accuracy with respect to the size of the basis set. We highlight two separate examples in applied electromagnetics and photonics: (i) homogenization of periodic structures, and (ii) numerical simulation of electromagnetic waves in slab geometries. Extensive numerical evidence and theoretical considerations show that Trefftz approximations can be applied much more broadly than is traditionally done: they are effective not only in physically homogeneous regions but also in complex inhomogeneous ones. Two mechanisms underlying the high accuracy of Trefftz approximations in such complex cases are pointed out. The first one is related to trigonometric interpolation and the second one – somewhat surprisingly – to well-posedness of random matrices.     

Formulation and analysis of fully-mixed methods for stress-assisted diffusion problems

This paper is devoted to the mathematical and numerical analysis of a mixed-mixed PDE system describing the stress-assisted diffusion of a solute into an elastic material. The equations of elastostatics are written in mixed form using stress, rotation and displacements, whereas the diffusion equation is also set in a mixed three-field form, solving for the solute concentration, for its gradient, and for the diffusive flux. This setting simplifies the treatment of the nonlinearity in the stress-assisted diffusion term. The analysis of existence and uniqueness of weak solutions to the coupled problem follows as combination of Schauder and Banach fixed-point theorems together with the Babu?ka–Brezzi and Lax–Milgram theories. Concerning numerical discretization, we propose two families of finite element methods, based on either PEERS or Arnold–Falk–Winther elements for elasticity, and a Raviart–Thomas and piecewise polynomial triplet approximating the mixed diffusion equation. We prove the well-posedness of the discrete problems, and derive optimal error bounds using a Strang inequality. We further confirm the accuracy and performance of our methods through computational tests.     

Radial Basis Function Methods for the Rosenau Equation and Other Higher Order PDEs

Meshfree methods based on radial basis function (RBF) approximation are of interest for numerical solution of partial differential equations because they are flexible with respect to the geometry of the computational domain, they can provide high order convergence, they are not more complicated for problems with many space dimensions and they allow for local refinement. The aim of this paper is to show that the solution of the Rosenau equation, as an example of an initial-boundary value problem with multiple boundary conditions, can be implemented using RBF approximation methods. We extend the fictitious point method and the resampling method to work in combination with an RBF collocation method. Both approaches are implemented in one and two space dimensions. The accuracy of the RBF fictitious point method is analyzed partly theoretically and partly numerically. The error estimates indicate that a high order of convergence can be achieved for the Rosenau equation. The numerical experiments show that both methods perform well. In the one-dimensional case, the accuracy of the RBF approaches is compared with that of the corresponding pseudospectral methods, showing similar or slightly better accuracy for the RBF methods. In the two-dimensional case, the Rosenau problem is solved both in a square domain and in an irregular domain with smooth boundary, to illustrate the capability of the RBF-based methods to handle irregular geometries.     

An Adaptive Time Step Procedure for a Parabolic Problem with Blow-up

In this paper we introduce and analyze a fully discrete approximation for a parabolic problem with a nonlinear boundary condition which implies that the solutions blow up in finite time. We use standard linear elements with mass lumping for the space variable. For the time discretization we write the problem in an equivalent form which is obtained by introducing an appropriate time re-scaling and then, we use explicit Runge-Kutta methods for this equivalent problem. In order to motivate our procedure we present it first in the case of a simple ordinary differential equation and show how the blow up time is approximated in this case. We obtain necessary and sufficient conditions for the blow-up of the numerical solution and prove that the numerical blow-up time converges to the continuous one. We also study, for the explicit Euler approximation, the localization of blow-up points for the numerical scheme. Received October 4, 2001; revised March 27, 2002 Published online: July 8, 2002