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In this paper we combine the dual-mixed finite element method with a Dirichlet-to-Neumann mapping (given in terms of a boundary integral operator) to solve linear exterior transmission problems in the plane. As a model we consider a second order elliptic equation in divergence form coupled with the Laplace equation in the exterior unbounded region. We show that the resulting mixed variational formulation and an associated discrete scheme using Raviart-Thomas spaces are well posed, and derive the...
A new finite element, which is continuously differentiable,
but only piecewise quadratic
polynomials on a type of uniform triangulations, is introduced.
We construct a local basis which
does not involve nodal values nor derivatives.
Different from the traditional finite elements, we have to
construct a special, averaging operator
which is stable and preserves quadratic polynomials.
We show the optimal order of approximation
of the finite element in interpolation, and in solving
the biharmonic...
We present a finite volume method based on the integration of the Laplace equation on both the cells of a primal almost arbitrary two-dimensional mesh and those of a dual mesh obtained by joining the centers of the cells of the primal mesh. The key ingredient is the definition of discrete gradient and divergence operators verifying a discrete Green formula. This method generalizes an existing finite volume method that requires “Voronoi-type” meshes. We show the equivalence of this finite volume...
We present a finite volume method based on the integration of the Laplace
equation on both the cells of a primal almost arbitrary two-dimensional
mesh and those of a
dual mesh obtained by joining the centers of the cells of the primal mesh.
The key ingredient is the definition of discrete gradient and divergence
operators verifying a discrete Green formula.
This method generalizes an existing finite volume method that
requires “Voronoi-type” meshes.
We show the equivalence of this finite volume...
In this article we develop a posteriori error estimates for second order linear elliptic problems with point sources in two- and three-dimensional domains. We prove a global upper bound and a local lower bound for the error measured in a weighted Sobolev space. The weight considered is a (positive) power of the distance to the support of the Dirac delta source term, and belongs to the Muckenhoupt’s class A2. The theory hinges on local approximation properties of either Clément or Scott–Zhang interpolation...
In this paper we combine the dual-mixed finite element method with a Dirichlet-to-Neumann mapping
(given in terms of a boundary integral operator) to solve linear exterior transmission problems in
the plane. As a model we consider a second order elliptic equation in divergence form coupled with
the Laplace equation in the exterior unbounded region. We show that the resulting mixed variational
formulation and an associated discrete scheme using Raviart-Thomas spaces are well posed, and derive
the...
In this paper, a new a posteriori error estimator for nonconforming convection diffusion
approximation problem, which relies on the small discrete problems solution in stars, has
been established. It is equivalent to the energy error up to data oscillation without any
saturation assumption nor comparison with residual estimator
Domain decomposition techniques provide a flexible tool for the numerical approximation of partial differential equations. Here, we consider mortar techniques for quadratic finite elements in 3D with different Lagrange multiplier spaces. In particular, we focus on Lagrange multiplier spaces which yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces in case of hexahedral triangulations. As a result, standard efficient iterative solvers as...
Domain decomposition techniques provide a flexible tool for the numerical
approximation of partial differential equations. Here, we consider
mortar techniques for quadratic finite elements in 3D with
different Lagrange multiplier spaces.
In particular, we
focus on Lagrange multiplier spaces
which yield optimal discretization
schemes and a locally supported basis for the associated
constrained mortar spaces in case
of hexahedral triangulations. As a result,
standard efficient iterative solvers...
A recovery-based a posteriori error estimator for the generalized Stokes problem is established based on the stabilized (linear/constant) finite element method. The reliability and efficiency of the error estimator are shown. Through theoretical analysis and numerical tests, it is revealed that the estimator is useful and efficient for the generalized Stokes problem.
In this paper we develop a residual based a posteriori error analysis for an augmented mixed finite element method applied to the problem of linear elasticity in the plane. More precisely, we derive a reliable and efficient a posteriori error estimator for the case of pure Dirichlet boundary conditions. In addition, several numerical experiments confirming the theoretical properties of the estimator, and illustrating the capability of the corresponding adaptive algorithm to localize the singularities...
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