RENORMALIZED SOLUTIONS OF ELLIPTIC EQUATIONS WITH ROBIN BOUNDARY CONDITIONS

In the present paper, we consider elliptic equations with nonlinear and nonlao mogeneous Robin boundary conditions of the type {-div（B（x,u）△u） = f in Ω,u=0 on Гo, B（x,u）Vu·n^-＋γ（x）h（u） = 9 on Г1,where f and g are the element of L^1（Ω） and L^1（Г1）, respectively. We define a notion of renormalized solution and we prove the existence of a solution. Under additionM assumptions on the matrix field B we show that the renormalized solution is unique.

The Asymptotics of the Two-Dimensional Wave Equation for a General Multi-Connected Vibrating Membrane with Piecewise Smooth Robin Boundary Conditions

The asymptotic expansion for small |t| of the trace of the wave kernel , where and are the eigenvalues of the negative Laplacian in the (x 2,x 2)-plane, is studied for a multi-connected vibrating membrane ? in R 2 surrounded by simply connected bounded domains ? j with smooth boundaries ?? j (j = 1, ..., n), where a finite number of piecewise smooth Robin boundary conditions on the piecewise smooth components Γ i (i = 1+k j?1, ..., k j ) of the boundaries ?? j are considered, such that and k 0 = 0. The basic problem is to extract information on the geometry of ? using the wave equation approach. Some geometric quantities of ? (e. g. the area of ?, the total lengths of its boundary, the curvature of its boundary, the number of the holes of ?, etc.) are determined from the asymptotic expansion of the trace of the wave kernel for small |t|.

Asymptotic Expansions of the Heat Kernel of the Laplacian for General Annular Bounded Domains with Robin Boundary Conditions：Further Results

The asymptotic expansions of the trace of the heat kernel Θ(t) = Σ_(v =1)~∞exp(-tλ_v) for small positive t, where {λ_v} are the eigenvalues of the negative Laplacian-△_n = -Σ_(i = 1)~n(partial deriv/(partial deriv)x^i)~2 in R^n (n = 2 or 3), are studied for ageneral annular bounded domain Ω with a smooth inner boundary (partial deriv)Ω_1 and a smoothouter boundary (partial deriv)Ω_2, where a finite number of piecewise smooth Robin boundaryconditions (partial deriv/(partial deriv)n_j + γ_j)φ = 0 on the components Γ_j(j = 1, …, k) of(partial deriv)Ω_1 and on the components Γ_j(j = k + 1, …, m) of (partial deriv)Ω_2 areconsidered such that (partial deriv)Ω_1 = ∪_(j = 1)~kΓ_j and (partial deriv)Ω_2 = ∪_(j = k +1)~mΓ_j and where the coefficients γ_j(j = 1, …, m) are piecewise smooth positive functions. Someapplications of Θ(t) for an ideal gas enclosed in the general annular bounded domain Ω are given.Further results are also obtained.

A Least-Squares/Fictitious Domain Method for Linear Elliptic Problems with Robin Boundary Conditions

In this article,we discuss a least-squares/fictitious domain method for the solution of linear elliptic boundary value problems with Robin boundary conditions.LetΩandωbe two bounded domains of R d such thatω⊂Ω.For a linear elliptic problem inΩ\ωwith Robin boundary condition on the boundaryγofω,our goal here is to develop a fictitious domain method where one solves a variant of the original problem on the fullΩ,followed by a well-chosen correction overω.This method is of the virtual control type and relies on a least-squares formulation making the problem solvable by a conjugate gradient algorithm operating in a well chosen control space.Numerical results obtained when applying our method to the solution of two-dimensional elliptic and parabolic problems are given;they suggest optimal order of convergence.

SUPERCONVERGENCE ANALYSIS OF THE POLYNOMIAL PRESERVING RECOVERY FOR ELLIPTIC PROBLEMS WITH ROBIN BOUNDARY CONDITIONS

We analyze the superconvergence property of the linear finite element method based on the polynomial preserving recovery(PPR)for Robin boundary elliptic problems on triangulartions.First,we improve the convergence rate between the finite element solution and the linear interpolation under the H1-norm by introducing a class of meshes satisfying the Condition(α,σ,μ).Then we prove the superconvergence of the recovered gradients post-processed by PPR and define an asymptotically exact a posteriori error estimator.Finally,numerical tests are provided to verify the theoretical findings.

Approximation of Derivative for a Singularly Perturbed Second-Order ODE of Robin Type with Discontinuous Convection Coefficient and Source Term

In this paper, a singularly perturbed Robin type boundary value problem for second-order ordinary differential equation with discontinuous convection coefficient and source term is considered. A robust-layer-resolving numerical method is proposed. An e-uniform global error estimate for the numerical solution and also to the numerical derivative are established. Numerical results are presented, which are in agreement with the theoretical predictions.

A WEAK GALERKIN MIXED FINITE ELEMENT METHOD FOR SECOND-ORDER ELLIPTIC EQUATIONS WITH ROBIN BOUNDARY CONDITIONS

In this paper, we present a weak Galerkin （WG） mixed finite element method for solving the second-order elliptic equations with Robin boundary conditions. Stability and a priori error estimates for the coupled method are derived. We present the optimal order error estimate for the WG-MFEM approximations in a norm that is related to the L^2 for the flux and H1 for the scalar function. Also an optimal order error estimate in L^2 is derived for the scalar approximation by using a duality argument. A series of numerical experiments is presented that verify our theoretical results.

Flow characteristics in a parallel-plate porous channel under convective boundary conditions and triple diffusion for the non-Darcy porous matrix

This article inspects the effect of triple diffusion in a vertical conduit encapsulated with porous matrix and subjected to third kind boundary conditions.Third kind boundary condition is a combination of Dirichlet and Neumann boundary conditions which specifies a linear combination of function and its derivative values on the boundary.Homogeneous chemical reaction along with viscous and Darcy dissipation effects are included.Adapting the Boussinesq approximation,the soultal buoyancy effects due to concentration gradients of the dispersed components are taken into account.Applying suitable transformations,the conservation equations are reduced into dimensionless form and the dimensionless parameters evolved are thermal Grashof number (0≤Λ_(1)≤20),solutal Grashof number(for species 1 and 2,0≤Λ_(2);Λ_(3)≤20),porous (2≤σ≤8) and inertial parameters (0≤Ι≤6),Biot numbers(at the left and right walls,1≤Bi_(1);Bi_(2)≤10),Brinkman number (0≤Br≤1),Schmidt numbers (0≤Sc_(1);Sc_(2)≤6),Soret numbers (Sr_(1) =Sr_(2) =1) and temperature difference ratio (R_(T) = 1).Adopting perturbation technique,the analytical solutions which are applicable only when the Brinkman number is less than one is appraised.However for any values of the Brinkman number,Runge-Kutta shooting method is operated.The impact of selected parameters on the momentum,heat and dual species concentration fields are presented in the form of pictures.The solutions computed by numerical method are justified by comparing with the analytical method.The numerical and analytical solutions are equal in the absence of Darcy and viscous dissipations and the discrepancy advances as the Brinkman number expands.Further the solutions obtained are also justified by comparing the results with Zanchini[1]in the absence of chemical reaction for clear fluid.The thermal field is augmented with the Brinkman number for symmetric and asymmetric Biot numbers.However the profiles are highly distinct at the cold plate for unequal Biot numbers in comparison with equal Biot numbers.The conclusions are admissible to materials processing and chemical transport phenomena.

Homogenization of Elliptic Problems with Quadratic Growth and Nonhomogenous Robin Conditions in Perforated Domains

This paper deals with the homogenization of a class of nonlinear elliptic problems with quadratic growth in a periodically perforated domain. The authors prescribe a Dirichlet condition on the exterior boundary and a nonhomogeneous nonlinear Robin condition on the boundary of the holes. The main difficulty, when passing to the limit, is that the solution of the problems converges neither strongly in L^2(Ω) nor almost everywhere in Ω. A new convergence result involving nonlinear functions provides suitable weak convergence results which permit passing to the limit without using any extension operator.Consequently, using a corrector result proved in [Chourabi, I. and Donato, P., Homogenization and correctors of a class of elliptic problems in perforated domains, Asymptotic Analysis, 92(1), 2015, 1–43, DOI: 10.3233/ASY-151288], the authors describe the limit problem, presenting a limit nonlinearity which is different for the two cases, that of a Neumann datum with a nonzero average and with a zero average.

Existence and Uniqueness of Almost Periodic Solution for a Mathematical Model of Tumor Growth

This article is concerned with a mathematical model of tumor growth governed by 2nd order diffusion equation . The source of mitotic inhibitor is almost periodic and time-dependent within the tissue. The system is set up with the initial condition C(r, 0) = C0(r) and Robin type inhomogeneous boundary condition . Under certain conditions we show that there exists a unique solution for this model which is almost periodic.

AN EXISTENCE－UNIQUENESS RESULT FOR SECOND ORDER NONLINEAR DIFEERENTIAL EQUATIONSWITH ROBIN BOUNDARY CONDITION

An existence-uniqueness result is given for second order nonlinear differential equations with Robin boundary conditionwhere αi, βi,(i =1,2), α and b are all constants.And the resonant points of this problemare also evaluated.AMS(MOS) Subject classifications 34B15

LINEARIZATION OF A NONLINEAR PERIODIC BOUNDARY CONDITION RELATED TO CORROSION MODELING

We study galvanic currents on a heterogeneous surface. In electrochemistry, the oxidation-reduction reaction producing the current is commonly modeled by a nonlinear elliptic boundary value problem. The boundary condition is of exponential type with periodically varying parameters. We construct an approximation by first homogenizing the problem, and then linearizing about the homogenized solution. This approximation is far more accurate than both previous approximations or direct linearization. We establish convergence estimates for both the two and three-dimensional case and provide two-dimensional numerical experiments.

Efficient Chebyshev Spectral Method for Solving Linear Elliptic PDEs Using Quasi-Inverse Technique

We present a systematic and efficient Chebyshev spectral method using quasiinverse technique to directly solve the second order equation with the homogeneous Robin boundary conditions and the fourth order equation with the first and second boundary conditions.The key to the efficiency of the method is to multiply quasiinverse matrix on both sides of discrete systems,which leads to band structure systems.We can obtain high order accuracy with less computational cost.For multi-dimensional and more complicated linear elliptic PDEs,the advantage of this methodology is obvious.Numerical results indicate that the spectral accuracy is achieved and the proposed method is very efficient for 2-D high order problems.