Crank-Nicolson-Galerkin approximation of the periodic solutions of weakly nonlinear parabolic equations
Curved elements in a mixed finite element method close to the equilibrium model
Curved triangular finite -elements
Curved triangular -elements which can be pieced together with the generalized Bell’s -elements are constructed. They are applied to solving the Dirichlet problem of an elliptic equation of the order in a domain with a smooth boundary by the finite element method. The effect of numerical integration is studied, sufficient conditions for the existence and uniqueness of the approximate solution are presented and the rate of convergence is estimated. The rate of convergence is the same as in the...
Cusp forms and Hecke groups.
Daubechies wavelets on intervals with application to BVPs
In this paper, Daubechies wavelets on intervals are investigated. An analytic technique for evaluating various types of integrals containing the scaling functions is proposed; they are compared with classical techniques. Finally, these results are applied to two-point boundary value problems.
Defect correction and a posteriori error estimation of Petrov-Galerkin methods for nonlinear Volterra integro-differential equations
We present two defect correction schemes to accelerate the Petrov-Galerkin finite element methods [19] for nonlinear Volterra integro-differential equations. Using asymptotic expansions of the errors, we show that the defect correction schemes can yield higher order approximations to either the exact solution or its derivative. One of these schemes even does not impose any extra regularity requirement on the exact solution. As by-products, all of these higher order numerical methods can also be...
Defect correction methods for convection dominated convection-diffusion problems
Der Einfluß von Randsingularitäten beim Ritzschen Verfahren. - Solution Effects of the Ritz Method.
Dérivabilité de l'erreur par rapport à la triangulation dans les méthodes d'éléments finis
Determination of the initial stress tensor from deformation of underground opening in excavation process
A method for the detection of the initial stress tensor is proposed. The method is based on measuring distances between pairs of points located on the wall of underground opening in the excavation process. This methods is based on solving twelve auxiliary problems in the theory of elasticity with force boundary conditions, which is done using the least squares method. The optimal location of the pairs of points on the wall of underground openings is studied. The pairs must be located so that the...
Determination of the initial stress tensor from deformation of underground opening -- theoretical background and applications
In this paper a method for the detection of initial stress tensor is proposed. The method is based on measuring distances between some pairs of points located on the wall of underground opening in the excavation process. This methods is based on the solution of eighteen auxiliary problems in the theory of elasticity with force boundary conditions. The optimal location of the pairs of points on the wall of underground work is studied. The pairs must be located so that the condition number of a certain...
Determination of the Thickness and Composition Profiles for a Film of Binary Mixture on a Solid Substrate
We determine the steady-state structures that result from liquid-liquid demixing in a free surface film of binary liquid on a solid substrate. The considered model corresponds to the static limit of the diffuse interface theory describing the phase separation process for a binary liquid (model-H), when supplemented by boundary conditions at the free surface and taking the influence of the solid substrate into account. The resulting variational problem...
Die Methode der finiten Elemente zur Lösung von elliptischen Randwertaufgaben.
Diffpack technical summary
Diffusion and propagation problems in some ramified domains with a fractal boundary
This paper is devoted to some elliptic boundary value problems in a self-similar ramified domain of with a fractal boundary. Both the Laplace and Helmholtz equations are studied. A generalized Neumann boundary condition is imposed on the fractal boundary. Sobolev spaces on this domain are studied. In particular, extension and trace results are obtained. These results enable the investigation of the variational formulation of the above mentioned boundary value problems. Next, for homogeneous...
Direct and Inverse Error Estimates for Finite Elements With Mesh Refinements.
Dirichlet-Neumann alternating algorithm for an exterior anisotropic quasilinear elliptic problem
In this paper, by the Kirchhoff transformation, a Dirichlet-Neumann (D-N) alternating algorithm which is a non-overlapping domain decomposition method based on natural boundary reduction is discussed for solving exterior anisotropic quasilinear problems with circular artificial boundary. By the principle of the natural boundary reduction, we obtain natural integral equation for the anisotropic quasilinear problems on circular artificial boundaries and construct the algorithm and analyze its convergence....
Discontinuous Galerkin and the Crouzeix–Raviart element : application to elasticity
We propose a discontinuous Galerkin method for linear elasticity, based on discontinuous piecewise linear approximation of the displacements. We show optimal order a priori error estimates, uniform in the incompressible limit, and thus locking is avoided. The discontinuous Galerkin method is closely related to the non-conforming Crouzeix–Raviart (CR) element, which in fact is obtained when one of the stabilizing parameters tends to infinity. In the case of the elasticity operator, for which the...
Discontinuous Galerkin and the Crouzeix–Raviart element: Application to elasticity
We propose a discontinuous Galerkin method for linear elasticity, based on discontinuous piecewise linear approximation of the displacements. We show optimal order a priori error estimates, uniform in the incompressible limit, and thus locking is avoided. The discontinuous Galerkin method is closely related to the non-conforming Crouzeix–Raviart (CR) element, which in fact is obtained when one of the stabilizing parameters tends to infinity. In the case of the elasticity operator, for...
Discontinuous Galerkin methods for problems with Dirac delta source∗
In this article we study discontinuous Galerkin finite element discretizations of linear second-order elliptic partial differential equations with Dirac delta right-hand side. In particular, assuming that the underlying computational mesh is quasi-uniform, we derive an a priori bound on the error measured in terms of the L2-norm. Additionally, we develop residual-based a posteriori error estimators that can be used within an adaptive mesh refinement ...