Nonhomogeneous Dirichlet Boundary Conditions Research Articles

High-order implicit Galerkin–Legendre spectral method for the two-dimensional Schrödinger equation

In this paper, we propose Galerkin–Legendre spectral method with implicit Runge-Kutta method for solving the unsteady two-dimensional Schrödinger equation with nonhomogeneous Dirichlet boundary conditions and initial condition. We apply a Galerkin–Legendre spectral method for discretizing spatial derivatives, and then employ the implicit Runge–Kutta method for the time integration of the resulting linear first-order system of ordinary differential equations in complex domain. We derive the spectral rate of convergence for the proposed method in the L2-norm for the semidiscrete formulation. Numerical experiments show our formulation have high-order accuracy.

A priori and a posteriori error analyses of an HDG method for the Brinkman problem

In this paper we introduce and analyze a hybridizable discontinuous Galerkin (HDG) method for the linear Brinkman model of porous media flow in two and three dimensions and with non-homogeneous Dirichlet boundary conditions. We consider a fully-mixed formulation in which the main unknowns are given by the pseudostress, the velocity and the trace of the velocity, whereas the pressure is easily recovered through a simple postprocessing. We show that the corresponding continuous and discrete schemes are well-posed. In particular, we use the projection-based error analysis in order to derive a priori error estimates. Furthermore, we develop a reliable and efficient residual-based a posteriori error estimator, and propose the associated adaptive algorithm for our HDG approximation. Finally, several numerical results illustrating the performance of the method, confirming the theoretical properties of the estimator and showing the expected behavior of the adaptive refinements are presented.

The existence of weak solutions for steady flows of electrorheological fluids with nonhomogeneous Dirichlet boundary condition

In this paper, we show the existence of weak solutions for steady flows of electrorheological fluids with nonhomogeneous Dirichlet boundary condition under the condition p(x)>2nn+2, n=2,3. In particular, we improve a priori estimate for weak solutions to the problem with minp(x)<2, which is an improvement even for constant p.

Space–time Legendre–Gauss–Lobatto collocation method for two-dimensional generalized sine-Gordon equation

In this paper, we present a space–time Legendre–Gauss–Lobatto (LGL) collocation method for solving the generalized two-dimensional sine-Gordon equation with nonhomogeneous Dirichlet boundary conditions. The proposed method is based on the LGL collocation method to discretize in space, then use the LGL collocation method or block LGL collocation method to discretize in time. Our formulation has high-order accuracy in both space and time. We introduce a new H1 projection for the error analysis of collocation method for nonhomogeneous Dirichlet boundary conditions. This projection differs from the classical H1 projection. The new H1 projection is identically equal to Lagrange interpolation on boundary. The approximation properties of the new H1 projection are obtain. We derive error bounds in both discrete L2 and H1 norms for the (spatially) semidiscrete formulation, the analysis is based on new H1 projection results.

A Posteriori Error Estimator for a Weak Galerkin Finite Element Solution of the Stokes Problem

AbstractA robust residual-based a posteriori error estimator is proposed for a weak Galerkin finite element method for the Stokes problem in two and three dimensions. The estimator consists of two terms, where the first term characterises the difference between the L2-projection of the velocity approximation on the element interfaces and the corresponding numerical trace, and the second is related to the jump of the velocity approximation between the adjacent elements. We show that the estimator is reliable and efficient through two estimates of global upper and global lower bounds, up to two data oscillation terms caused by the source term and the nonhomogeneous Dirichlet boundary condition. The estimator is also robust in the sense that the constant factors in the upper and lower bounds are independent of the viscosity coefficient. Numerical results are provided to verify the theoretical results.

Nonhomogeneous boundary value problems for the complex Ginzburg–Landau equation posed on a finite interval

This paper is concerned with the well-posedness of the complex Ginzburg–Landau equation posed on a finite interval (0, L) with nonhomogeneous Dirichlet boundary conditions. When the initial datum belongs to () and the boundary data belong to some subspace of , the complex Ginzburg–Landau equation is shown to be globally well-posed.

A finite volume scheme for boundary-driven convection–diffusion equations with relative entropy structure

We propose a finite volume scheme for a class of nonlinear parabolic equations endowed with non-homogeneous Dirichlet boundary conditions and which admit relative entropy functionals. For this kind of models including the porous media equations, Fokker-Planck equations for plasma physics or dumbbell models for polymer flows, it has been proved that the transient solution converges to a steady-state when time goes to infinity. The present scheme is built from the resolution of the stationary equation in order to preserve steady-states and natural Lyapunov functionals which provide a satisfying long-time behavior. After describing the numerical scheme, we present several numerical results which confirm the accuracy and underline the efficiency to preserve the large-time asymptotic.

Convergence order estimates of the local discontinuous Galerkin method for instationary Darcy flow

We present a new a priori stability and convergence analysis for the local discontinuous Galerkin method applied to the instationary Darcy problem formulated on a d‐dimensional polytope with nonhomogeneous Neumann and Dirichlet boundary conditions. In addition to including a spatially and temporally varying permeability tensor into all estimates, the utilized analysis technique produces a convergence order result for the primary and the flux variables. The only stabilization in the proposed scheme is represented by a penalty term in the primary unknown, and our analysis provides some insights into the role played by this particular choice of stabilization. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1374–1394, 2017

A mixed virtual element method for the Brinkman problem

In this paper, we introduce and analyze a mixed virtual element method (mixed-VEM) for the two-dimensional Brinkman model of porous media flow with non-homogeneous Dirichlet boundary conditions. More precisely, we employ a dual-mixed formulation in which the only unknown is given by the pseudostress, whereas the velocity and pressure are computed via postprocessing formulae. We first recall the corresponding variational formulation, and then summarize the main mixed-VEM ingredients that are required for our discrete analysis. In particular, in order to define a calculable discrete bilinear form, whose continuous version involves deviatoric tensors, we propose two well-known alternatives for the local projector onto a suitable polynomial subspace, which allows the explicit integration of these terms. Next, we show that the global discrete bilinear form satisfies the hypotheses required by the Lax–Milgram lemma. In this way, we conclude the well-posedness of our mixed-VEM scheme and derive the associated a priori error estimates for the virtual solution as well as for the fully computable projection of it. Furthermore, we also introduce a second element-by-element postprocessing formula for the pseudostress, which yields an optimally convergent approximation of this unknown with respect to the broken ℍ(div)-norm. Finally, several numerical results illustrating the good performance of the method and confirming the theoretical rates of convergence are presented.

Totally analytical closure of space filtered Navier–Stokes for arbitrary Reynolds number: Part III. aFNS theory validation

ABSTRACTFor arbitrary Reynolds number Re, aFNS theory O(1; δ2; δ3) significance scaled state variable achieves analytical closure of rigorously space filtered thermal, multi-species Navier–Stokes (NS) conservation principle partial differential equation (PDE) system well-posed on bounded domains (Part I). Validation of boundary commutation error (BCE) and nonhomogeneous Dirichlet boundary condition (DBC) resolution strategies results in ADh-GWSh-BCEh-DBCh algorithm coupling with aFNS theory optimal Galerkin GWSh + θTS CFD algorithm (Part II), including accuracy/convergence assessments. For Reynolds numbers ranging E+02 <Re < 2E+04, aFNS theory fully coupled k = 1 basis Galerkin CFD code generated a posteriori data enable theory quantitative validation. For assured laminar Re range, NS no-slip BC reduced and coupled aFNS theory code predicted quasi-steady thermal-velocity transition to periodic unsteady is validated via linear stability predicted critical Re. For Re exceeding assured laminar, theory coupled CFD data quantify spatial filtering annihilation of (laminar) NS predicted large wave number spectral content. For sufficiently large Re, coupled CFD code a posteriori data document first principles prediction of unsteady periodic wall attached velocity profile transitioning-from-laminar, then separation, fully turbulent profile reattachment followed by relaminarization. These large Re CFD data as well enable validation of perturbation theory O(1; δ2; δ3) significance scaled state variable, ADh-GWSh-DBCh algorithm prediction of O(δ2) state variable non-homogeneous Dirichlet BC DOF data and in concert thoroughly quantify theory generated O(δ2; δ3) state variable distributions.

Totally analytical closure of space filtered Navier–Stokes for arbitrary Reynolds number: Part II. Resolution algorithms, validations

ABSTRACTFor uniform measure δ Gaussian filter, Part I derives the totally analytical aFNS theory closing rigorously space filtered Navier–Stokes (NS) partial differential equation (PDE) system absent a Reynolds number (Re) assumption. aFNS theory state variable is scaled O(1; δ2; δ3) via classic fluid mechanics perturbation theory which also identifies the O(δ2) elliptic PDE system. Filter penetration of domain bounding surfaces requires O(1) PDE system inclusion of boundary commutation error (BCE) integrals. For the O(1; δ2), PDE system to be bounded domain well-posed requires derivation of domain encompassing nonhomogeneous Dirichlet boundary conditions (DBC). Resolution of BCE and DBC requirements is theorized via O(δ4) approximate deconvolution (AD) Galerkin differential definition weak form algorithms. Amenable to any space-time discretization, detailed is aFNS theory insertion in the optimal Galerkin weak form CFD algorithm, finite element linear tensor product basis implemented. Coupled Galerkin CFD/BCE/DBC code a posteriori data reported herein validate theory resolution algorithms including accuracy/convergence assessments.

Totally analytical closure of space filtered Navier–Stokes for arbitrary Reynolds number: Part I. Theory, resolutions

ABSTRACTRigorously space filtering the thermal, multispecies Navier–Stokes (NS) conservation principle partial differential equation (PDE) system embeds a priori undefined tensor and vector quadruples. Large eddy simulation (LES) computational fluid dynamics algorithm resolutions replace the tensor quadruple with a single tensor then secures closure through “physics-based” modeling, assuming the velocity field is turbulent, i.e., the Reynolds number (Re) is large. In complete distinction, a totally analytical closure is derived for the rigorously generated tensor/vector quadruples, achieved totally absent any modeling component or Re assumption. For Gaussian filter of uniform measure δ, derived analytical filtered Navier–Stokes (aFNS) theory PDE system state variable is significance scaled O(1; δ2; δ3) through classic fluid mechanics perturbation theory. That uniform measure δ filter penetrates domain boundaries requires O(1) resolved scale PDE system inclusion of boundary commutation error (BCE) integrals, (unfiltered) NS state variable extension in the sense of distributions, and domain enlargement to encompass all surfaces with Dirichlet boundary condition (DBC) specification. Theory-derived O(δ2) resolved–unresolved scale interaction PDE system, also the O(1) system, is rendered bounded domain, well posed through a priori identification of O(1; δ2) state variable nonhomogeneous DBCs. BCE and DBC resolution algorithm derivations use O(δ4) approximate deconvolution (AD) differential definition Galerkin weak forms. Theory analytically derived unresolved scale O(δ3) state variable annihilates discretization-induced O(h2) dispersion error at unresolved scale threshold δ, h the mesh measure. Net is an analytical theory closing rigorously space-filtered NS exhibiting potential for first principles prediction of viscous laminar–turbulent transition, separation, and relaminarization.

Analysis of average Nusselt numbers at various zones for heat flow visualizations during natural convection within enclosures (square vs triangular) involving discrete heaters

Finite element analysis of average Nusselt numbers during natural convection within the triangular (type 1 and type 2) and square enclosures with various discrete heater configurations along the side walls (cases 1–4) is carried out for various elements (24×24, 28×28, 32×32 and 34×34) with Pr=0.7 fluids. The adaptive finite elements have been used for all the grid sizes as they are useful in accurately predicting the average Nusselt number. The average Nusselt numbers further demonstrate the heat balance within the enclosures. The fractional error in the heat balance (ε) has been evaluated for all the cases and various elements. It was found that the fractional error is lesser for the elements corresponding to 32×32 (ε≤0.109) and 34×34 (ε≤0.105). The fractional error is mostly due to the sharp variation of the heat transfer rate at the hot–cold interface along the side walls and that is also correctly predicted by the local Nusselt number profiles for various elements. The present work on the discrete heating along the side walls also involve the derivation of various non-homogeneous Dirichlet boundary conditions for heatfunctions (Π) at the hot–cold junctions. The heatfunctions are inherently related to the average Nusselt number. It was finally concluded that the implementation of the heatfunction boundary conditions based on the average Nusselt numbers successfully predicts the heat flow visualizations via heatlines.

A priori and a posteriori error analyses of a pseudostress-based mixed formulation for linear elasticity

In this paper we present the a priori and a posteriori error analyses of a non-standard mixed finite element method for the linear elasticity problem with non-homogeneous Dirichlet boundary conditions. More precisely, the approach introduced here is based on a simplified interpretation of the pseudostress–displacement formulation originally proposed in Arnold and Falk (1988), which does not require symmetric tensor spaces in the finite element discretization. In addition, physical quantities such as the stress, the strain tensor of small deformations, and the rotation, are computed through a simple postprocessing in terms of the pseudostress variable. Furthermore, we also introduce a second element-by-element postprocessing formula for the stress, which yields an optimally convergent approximation of this unknown with respect to the broken H(div)-norm. We apply the classical Babuška–Brezzi theory to prove that the corresponding continuous and discrete schemes are well-posed. In particular, Raviart–Thomas spaces of order k≥0 for the pseudostress and piecewise polynomials of degree ≤k for the displacement can be utilized. Moreover, we remark that in the 3D case the number of unknowns behaves approximately as 9 times the number of elements (tetrahedra) of the triangulation when k=0. This factor increases to 12.5 when one uses the classical PEERS. Next, we derive a reliable and efficient residual-based a posteriori error estimator for the mixed finite element scheme. Finally, several numerical results illustrating the performance of the method, confirming the theoretical properties of the estimator, and showing the expected behaviour of the associated adaptive algorithm, are provided.

The vanishing diffusivity limit for the 2-D Boussinesq equations with boundary effect

In this paper, we tackle the issue of the vanishing diffusivity limit of the 2-D incompressible Boussinesq equations in the half plane. Our main purpose is to study the boundary layer effect and the convergence rates as the thermal diffusion parameter ϵ goes to zero. Under the homogeneous Dirichlet boundary condition of velocity and the nonhomogeneous Dirichlet boundary condition in the x-direction for temperature, we show that the boundary layer thickness is of the order O(ϵβ) with (0<β<23). In contrast with Jiang et al. (2011), the BL-thickness we got is thinner than that in Jiang et al. (2011). Moreover, we prove that as diffusivity tends to zero, the solutions of the fully viscous equations converge strongly to those of zero diffusion equations. We also obtain the convergence rates of the vanishing diffusivity limit.

Open boundary conditions for ISPH and their application to micro-flow

Open boundary conditions for incompressible Smoothed Particle Hydrodynamics (ISPH) are rare. For stable simulations with open boundary conditions, one needs to specify all boundary conditions correctly in the pressure force as well as in the linear equation system for pressure calculation. Especially for homogeneous or non-homogeneous Dirichlet boundary conditions for pressure there exist several possibilities but only a few lead to stable results. However, this isn't trivial for open boundary conditions. We introduce a new approach for open boundary conditions for ISPH to enable stable simulations. In contrast to existing models for weakly-compressible SPH, we can specify open pressure boundary conditions because in ISPH, pressure can be calculated independently of the density. The presented approach is based on the mirror particle approach already introduced for solid wall boundary conditions. Here we divide the mirror axis in several segments with time-dependent positions.We validate the presented approach for the example of Poiseuille flow and flow around a cylinder at different Reynolds numbers and show that we get good agreement with references. Then, we demonstrate that the approach can be applied to free surface flows. Finally, we apply the new approach to micro-flow through a random porous medium with a different number of in- and outlets and demonstrate its benefits.

A Stochastic Conservation Law with Nonhomogeneous Dirichlet Boundary Conditions

This paper discusses the initial-boundary value problem (with a nonhomogeneous boundary condition) for a multi-dimensional scalar first-order conservation law with a multiplicative noise. One introduces a notion of kinetic formulations in which the kinetic defect measures on the boundary of a domain are truncated. In such a kinetic formulation, one obtains a result of uniqueness and existence. The unique solution is the limit of the solution of the stochastic parabolic approximation.

Justification of lubrication approximation: An application to fluid/solid interactions

We consider the stationary Stokes problem in a three-dimensional fluid domain $\mathcal F$ with non-homogeneous Dirichlet boundary conditions. We assume that this fluid domain is the complement of a bounded obstacle $\mathcal B$ in a bounded or an exterior smooth container $\Omega$. We compute sharp asymptotics of the solution to the Stokes problem when the distance between the obstacle and the container boundary is small.

A Model for Selection of Eyespots on Butterfly Wings.

Unsolved ProblemThe development of eyespots on the wing surface of butterflies of the family Nympalidae is one of the most studied examples of biological pattern formation.However, little is known about the mechanism that determines the number and precise locations of eyespots on the wing. Eyespots develop around signaling centers, called foci, that are located equidistant from wing veins along the midline of a wing cell (an area bounded by veins). A fundamental question that remains unsolved is, why a certain wing cell develops an eyespot, while other wing cells do not.Key Idea and ModelWe illustrate that the key to understanding focus point selection may be in the venation system of the wing disc. Our main hypothesis is that changes in morphogen concentration along the proximal boundary veins of wing cells govern focus point selection. Based on previous studies, we focus on a spatially two-dimensional reaction-diffusion system model posed in the interior of each wing cell that describes the formation of focus points. Using finite element based numerical simulations, we demonstrate that variation in the proximal boundary condition is sufficient to robustly select whether an eyespot focus point forms in otherwise identical wing cells. We also illustrate that this behavior is robust to small perturbations in the parameters and geometry and moderate levels of noise. Hence, we suggest that an anterior-posterior pattern of morphogen concentration along the proximal vein may be the main determinant of the distribution of focus points on the wing surface. In order to complete our model, we propose a two stage reaction-diffusion system model, in which an one-dimensional surface reaction-diffusion system, posed on the proximal vein, generates the morphogen concentrations that act as non-homogeneous Dirichlet (i.e., fixed) boundary conditions for the two-dimensional reaction-diffusion model posed in the wing cells. The two-stage model appears capable of generating focus point distributions observed in nature.ResultWe therefore conclude that changes in the proximal boundary conditions are sufficient to explain the empirically observed distribution of eyespot focus points on the entire wing surface. The model predicts, subject to experimental verification, that the source strength of the activator at the proximal boundary should be lower in wing cells in which focus points form than in those that lack focus points. The model suggests that the number and locations of eyespot foci on the wing disc could be largely controlled by two kinds of gradients along two different directions, that is, the first one is the gradient in spatially varying parameters such as the reaction rate along the anterior-posterior direction on the proximal boundary of the wing cells, and the second one is the gradient in source values of the activator along the veins in the proximal-distal direction of the wing cell.

Boussinesq system with non-homogeneous boundary conditions

A classical stationary Boussinesq system with non-homogeneous Dirichlet boundary conditions in a bounded domain Ω ⊂ R 3 is considered in this paper; included is the case of a possibly disconnected boundary. We prove existence of a weak, a strong and a very weak solution in L p -theory. Uniqueness of the very weak solution is proved under a small data assumption.