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Cell-centered and vertex-centered finite volume schemes for the Laplace equation with homogeneous Dirichlet boundary conditions are considered on a triangular mesh and on the Voronoi diagram associated to its vertices. A broken P1 function is constructed from the solutions of both schemes. When the domain is two-dimensional polygonal convex, it is shown that this reconstruction converges with second-order accuracy towards the exact solution in the L2 norm, under the sufficient condition that the...
Cell-centered and vertex-centered finite volume schemes for the Laplace equation
with homogeneous Dirichlet boundary conditions
are considered on a triangular mesh and on the Voronoi diagram associated to its vertices.
A broken P1 function is constructed from the solutions of both schemes.
When the domain is two-dimensional polygonal convex,
it is shown that this reconstruction
converges with second-order accuracy towards the exact solution in the L2 norm,
under the sufficient condition that the...
The solvability of time-harmonic Maxwell equations in the 3D-case with nonhomogeneous conductivities is considered by adapting Sobolev space technique and variational formulation of the problem in question. Moreover, a finite element approximation is presented in the 3D-case together with an error estimate in the energy norm. Some remarks are given to the 2D-problem arising from geophysics.
We consider the original DG method for solving the advection-reaction equations with arbitrary velocity in space dimensions. For triangulations satisfying the flow condition, we first prove that the optimal convergence rate is of order in the -norm if the method uses polynomials of order . Then, a very simple derivative recovery formula is given to produce an approximation to the derivative in the flow direction which superconverges with order . Further we consider a residual-based a posteriori...
We prove the optimal convergence of a discontinuous-Galerkin-based
immersed boundary method introduced earlier [Lew and Buscaglia, Int. J. Numer. Methods Eng.76 (2008) 427–454]. By switching to a discontinuous
Galerkin discretization near the boundary, this method overcomes the
suboptimal convergence rate that may arise in immersed boundary
methods when strongly imposing essential boundary conditions. We
consider a model Poisson's problem with homogeneous boundary
conditions over two-dimensional...
We prove the optimal convergence of a discontinuous-Galerkin-based
immersed boundary method introduced earlier [Lew and Buscaglia, Int. J. Numer. Methods Eng.76 (2008) 427–454]. By switching to a discontinuous
Galerkin discretization near the boundary, this method overcomes the
suboptimal convergence rate that may arise in immersed boundary
methods when strongly imposing essential boundary conditions. We
consider a model Poisson's problem with homogeneous boundary
conditions over two-dimensional...
We present an improved, near-optimal hp error estimate for a
non-conforming finite element method, called the mortar method (M0). We
also present a new hp mortaring technique, called the mortar method (MP),
and derive h, p and hp error estimates for it, in the presence of
quasiuniform and non-quasiuniform meshes. Our theoretical results,
augmented by the computational evidence we present, show that like (M0), (MP)
is also a viable mortaring technique for the hp method.
Finite element semidiscrete approximations on nonlinear dynamic
shallow shell models in considered. It is shown that the algorithm
leads to global, optimal rates of convergence. The result
presented in the paper improves upon the existing literature where the
rates of convergence were derived for small initial data only
[19].
In this short note we provide an optimal analysis of finite element convergence on meshes containing a so-called band of caps. These structures consist of a zig-zag arrangement of ‘degenerating’ triangles which violate the maximum angle condition. A necessary condition on the geometry of such a structure for various -convergence rates was previously given by Kučera. Here we prove that the condition is also sufficient, providing an optimal analysis of this special case of meshes. In the special...
We consider a finite element discretization by
the Taylor–Hood element for the stationary
Stokes and Navier–Stokes
equations with slip boundary condition. The slip boundary condition
is enforced pointwise for nodal values of the velocity in
boundary nodes. We prove optimal error estimates in the
H1 and L2 norms for the velocity and pressure respectively.
We consider the analysis and
numerical solution of a forward-backward boundary value problem.
We provide some motivation, prove existence and uniqueness in a function
class especially geared to the problem at hand, provide various energy
estimates, prove a priori error estimates for the Galerkin method,
and show the results of some numerical computations.
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