Total Flux Estimates for a Finite-Element Approximation of Elliptic Equations

An elliptic boundary-value problem on a domain with prescribedDirichlet data on _I is approximated using a finite-elementspace of approximation power h^K in the L² norm. It is shownthat the total flux across _I can be approximated with an errorof O(h^K) when is a curved domain in Rⁿ (n = 2 or 3) and isoparametricelements are used. When is a polyhedron, an O(h^2K–2)approximation is given. We use these results to study the finite-elementapproximation of elliptic equations when the prescribed boundarydata on _I is the total flux. Present address: School of Mathematical and Physical Sciences,University of Sussex, Brighton, Sussex BN1 9QH.     

A Finite-element Method for Solving Elliptic Equations with Neumann Data on a Curved Boundary Using Unfitted Meshes

This paper considers a finite-element approximation of a Poissonequation in a region with a curved boundary on which a Neumanncondition is prescribed. Piecewise linear and bilinear elementsare used on unfitted meshes with the region of integration beingreplaced by a polygonal approximation. It is shown, despitethe variational crimes, that the rate of convergence is stillorder (h) in the H¹ norm. Numerical examples show that the methodis easy to implement and that the predicted rate of convergenceis obtained. Supported by SERC postdoctoral fellowship RF/5830.     

The Dirichlet Problem for Quasilinear Elliptic Equations in Domains with Smooth Closed Edges

The solvability of the Dirichlet problem for quasilinear elliptic second-order equations of nondivergence form are studied in a domain whose boundary contains a conical point or an edge of an arbitrary codimension. Bibliography: 12 titles.     

NUMERICAL METHODS FOR MAXWELL＇S EQUATIONS IN INHOMOGENEOUS MEDIA WITH MATERIAL INTERFACES

In this paper, we will present some recent results on developing numerical methods for solving Maxwell‘s equations in inhomogeneous media with material interfaces. First,we will present a second order upwinding embedded boundary method - a Cartesian grid based finite difference method with special upwinding treatment near the material interfaces. Second, we will present a high order discontinuous spectral element with Dubinar orthogonal polynomials on triangles. Numerical results on electromagnetic scattering and photonic waveguide will be included.     

Cubic Finite Volume Methods for Second Order Elliptic Equations with Variable

In this paper, we present a finite volume framework for second order elliptic equations with variable coefficients based on cubic Hermite element. We prove the optimal H^1 norm error estimates. A numerical example is given at the end to show the feasibility of the method.     

Finite-Element Approximation of Elliptic Equations with a Neumann or Robin Condition on a Curved Boundary

This paper considers a finite-element approximation of a second-orderself adjoint elliptic equation in a region Rⁿ (with n=2 or 3)having a curved boundary on which a Neumann or Robin conditionis prescribed. If the finite-element space defined over , a union of elements, has approximation power h^kin the L² norm, and if the region of integration is approximatedby ^h with dist (, ^h)Ch^k, then it is shown that one retains optimalrates of convergence for the error in the H¹ and L² norms, whetherQ^h is fitted or unfitted , provided that the numerical integration scheme has sufficientaccuracy.     

二次变系数椭圆方程的三次有限体积元方法

In this paper, we present a finite volume framework for second order elliptic equations with variable coefficients based on cubic Hermite element. We prove the optimal H1 norm error estimates. A numerical example is given at the end to show the feasibility of the method.     

Uniform Convergence of Adaptive Multigrid Methods for Elliptic Problems and Maxwell's Equations

We consider the convergence theory of adaptive multigrid methods for second-order elliptic problems and Maxwell's equations. The multigrid algorithm only performs pointwise Gauss-Seidel relaxations on new degrees of freedom and their "immediate" neighbors. In the context of lowest order conforming finite element approximations, we present a unified proof for the convergence of adaptive multigrid V-cycle algorithms. The theory applies to any hierarchical tetrahedral meshes with uniformly bounded shape-regularity measures. The convergence rates for both problems are uniform with respect to the number of mesh levels and the number of degrees of freedom. We demonstrate our convergence theory by two numerical experiments.     

Solvability of Elliptic Differential Equations,Set in Three Habitats with Skewness Boundary Conditions at the Interfaces

In this work, we study an elliptic differential equation set in three habitats with skewness boundary conditions at the interfaces. It represents the linear stationary case of dispersal problems of population dynamics which incorporate responses at interfaces between the habitats. Existence, uniqueness and regularity of the solution of these problems are obtained in Hölder spaces under necessary and sufficient conditions on the data. Our techniques are based on the semigroup theory, the fractional powers of linear operators, the \(H^{\infty }\) functional calculus for sectorial operators in Banach spaces and some properties of real interpolation spaces.     

Non-Classical Elliptic Projections and $L^2$-Error Estimates for Galerkin Methods for Parabolic Integro-Differential Equations

In this paper we shall define a so-called "non-classical" elliptic projection associated with an integro-differential operator. The properties of this projection will be analyzed and used to obtain the optimal $L^2$ error estimates for the continuous and discrete time Galerkin procedures when applied to linear integro-differential equations of parabolic type.     

Development and Analysis of Higher Order Finite Volume Methods over Rectangles for Elliptic Equations

Currently used finite volume methods are essentially low order methods. In this paper, we present a systematic way to derive higher order finite volume schemes from higher order mixed finite element methods. Mostly for convenience but sometimes from necessity, our procedure starts from the hybridization of the mixed method. It then approximates the inner product of vector functions by an appropriate, critical quadrature rule; this allows the elimination of the flux and Lagrange multiplier parameters so as to obtain equations in the scalar variable, which will define the finite volume method. Following this derivation with different mixed finite element spaces leads to a variety of finite volume schemes. In particular, we restrict ourselves to finite volume methods posed over rectangular partitions and begin by studying an efficient second-order finite volume method based on the Brezzi–Douglas–Fortin–Marini space of index two. Then, we present a general global analysis of the difference between the solution of the underlying mixed finite element method and its related finite volume method. Then, we derive finite volume methods of all orders from the Raviart–Thomas two-dimensional rectangular elements; we also find finite volume methods to associate with BDFM ₂ three-dimensional rectangles. In each case, we obtain optimal error estimates for both the scalar variable and the recovered flux.     

Superconvergence of H 1-Galerkin Mixed Finite Element Methods for Second-Order Elliptic Equations

We derive superconvergence result for H ¹-Galerkin mixed finite element method for second-order elliptic equations over rectangular partitions. Compared to standard mixed finite element procedure, the method is not subject to the Ladyzhenskaya–Bab?ska–Brezzi (LBB) condition and the approximating finite element spaces are allowed to be of different polynomial degrees. Superconvergence estimate of order 𝒪(h ^k+3), where k ≥ 1 is the order of the approximating polynomials employed in the Raviart–Thomas elements, is established for the flux via a postprocessing technique.     

Schauder Estimates for Elliptic and Parabolic Equations

In this note the author gives an elementary and simple proof for the Schauder estimates for elliptic and parabolic equations. The proof also applies to nonlinear equations.     

Nonlinear Elliptic Equations with Mixed Singularities

We study non-variational degenerate elliptic equations with mixed singular structures, both at the set of critical points and on the ground touching points. No boundary data are imposed and singularities occur along an a priori unknown interior region. We prove that positive solutions have a universal modulus of continuity that does not depend on their infimum value. We further obtain sharp, quantitative regularity estimates for non-negative limiting solutions.     

Spectral Solutions of Self-adjoint Elliptic Problems with Immersed Interfaces

This paper describes a spectral representation of solutions of self-adjoint elliptic problems with immersed interfaces. The interface is assumed to be a simple non-self-intersecting closed curve that obeys some weak regularity conditions. The problem is decomposed into two problems, one with zero interface data and the other with zero exterior boundary data. The problem with zero interface data is solved by standard spectral methods. The problem with non-zero interface data is solved by introducing an interface space H _Γ(Ω) and constructing an orthonormal basis of this space. This basis is constructed using a special class of orthogonal eigenfunctions analogously to the methods used for standard trace spaces by Auchmuty (SIAM J. Math. Anal. 38, 894–915, 2006). Analytical and numerical approximations of these eigenfunctions are described and some simulations are presented.     

Rearrangement Optimization for Some Elliptic Equations

We investigate an optimization problem related to an elliptic (linear and nonlinear) boundary-value problem. The competing objects are elements of a rearrangement class generated by a fixed positive function. The popular case where the generator is a characteristic function is also considered. In this case, the method of the domain derivative is used to obtain a free boundary result. This research was initiated when the first author was visiting the Iran University of Science and Technology in Summer of 2005. Behrouz Emamizadeh acknowledges the partial financial support from the Petroleum Institute.     

Removable Singularities for Anisotropic Elliptic Equations

We study a class of quasi-linear elliptic equations with model representative \(\sum _{i=1}^{n}(|u_{x_{i}}|^{p_{i}-2}u_{x_{i}})_{x_{i}}=0\) , which solutions have singularities on a smooth manifold. We establish the condition for removability of singularity on a manifold for solutions of such equations.     

Liouville Type Theorems for Elliptic Equations with Gradient Terms

In this paper we obtain Liouville type theorems for nonnegative supersolutions of the elliptic problem ${-\Delta u + b(x)|\nabla u| = c(x)u}$ in exterior domains of ${\mathbb{R}^N}$ . We show that if lim ${{\rm inf}_{x \longrightarrow \infty} 4c(x) - b(x)^2 > 0}$ then no positive supersolutions can exist, provided the coefficients b and c verify a further restriction related to the fundamental solutions of the homogeneous problem. The weights b and c are allowed to be unbounded. As an application, we also consider supersolutions to the problems ${-\Delta u + b|x|^{\lambda}|{\nabla} u| = c|x|^{\mu} u^p}$ and ${-\Delta u + be^{\lambda |x|}|\nabla u| = ce^{\mu |x|}u^p}$ , where p > 0 and λ, μ ≥ 0, and obtain nonexistence results which are shown to be optimal.     

Comparison Principle for Elliptic Equations with Mixed Singular Nonlinearities

Potential Analysis - We deal with existence and uniqueness of positive solutions of an elliptic boundary value problem modeled by $$ left {begin {array}{ll} displaystyle -{Delta }_{p} u= frac...     

Existence Results for Quasilinear Elliptic Equations with Discontinuous Nonlinearities

The nonsmooth critical point theory is applied to prove the existence of solutions and multiple solutions of a quasilinear elliptic equation with discontinuous nonlinearities.