Green's function of a centered partial difference equation.
Making use of a line integral defined without use of the partition of unity, Green’s theorem is proved in the case of two-dimensional domains with a Lipschitz-continuous boundary for functions belonging to the Sobolev spaces
This contribution presents a general numerical method for computing lower and upper bound of the optimal constant in Friedrichs’ inequality. The standard Rayleigh-Ritz method is used for the lower bound and the method of is employed for the upper bound. Several numerical experiments show applicability and accuracy of this approach.
We derive a posteriori error estimates for singularly perturbed reaction–diffusion problems which yield a guaranteed upper bound on the discretization error and are fully and easily computable. Moreover, they are also locally efficient and robust in the sense that they represent local lower bounds for the actual error, up to a generic constant independent in particular of the reaction coefficient. We present our results in the framework of the vertex-centered finite volume method but their nature...
We deal with the a-posteriori estimation of the error for finite element solutions of nonstationary heat conduction problems with mixed boundary conditions on bounded polygonal domains. The a-posteriori error estimates are constucted by solving stationary “reconstruction” problems, obtained by replacing the time derivative of the exact solution by the time derivative of the finite element solution. The main result is that the reconstructed solutions, or reconstructions, are superconvergent approximations...
A method of characterizing all eigenvalues of a preconditioned discretized scalar diffusion operator with Dirichlet boundary conditions has been recently introduced in Gergelits, Mardal, Nielsen, and Strakoš (2019). Motivated by this paper, we offer a slightly different approach that extends the previous results in some directions. Namely, we provide bounds on all increasingly ordered eigenvalues of a general diffusion or elasticity operator with tensor data, discretized with the conforming finite...
In this work, we analyze hierarchic -finite element discretizations of the full, three-dimensional plate problem. Based on two-scale asymptotic expansion of the three-dimensional solution, we give specific mesh design principles for the -FEM which allow to resolve the three-dimensional boundary layer profiles at robust, exponential rate. We prove that, as the plate half-thickness tends to zero, the -discretization is consistent with the three-dimensional solution to any power of in the energy...
We propose and analyze a domain decomposition method on non-matching grids for partial differential equations with non-negative characteristic form. No weak or strong continuity of the finite element functions, their normal derivatives, or linear combinations of the two is imposed across the boundaries of the subdomains. Instead, we employ suitable bilinear forms defined on the common interfaces, typical of discontinuous Galerkin approximations. We prove an error bound which is optimal with respect...
A brief survey is given to show that harmonic averages enter in a natural way in the numerical solution of various variable coefficient problems, such as in elliptic and transport equations, also of singular perturbation types. Local Green’s functions used as test functions in the Petrov-Galerkin finite element method combined with harmonic averages can be very efficient and are related to exact difference schemes.
Hermite polynomial interpolation is investigated. Some approximation results are obtained. As an example, the Burgers equation on the whole line is considered. The stability and the convergence of proposed Hermite pseudospectral scheme are proved strictly. Numerical results are presented.
We study the approximation properties of some finite element subspaces of H(div;Ω) and H(curl;Ω) defined on hexahedral meshes in three dimensions. This work extends results previously obtained for quadrilateral H(div;Ω) finite elements and for quadrilateral scalar finite element spaces. The finite element spaces we consider are constructed starting from a given finite dimensional space of vector fields on the reference cube, which is then transformed to a space of vector fields on a hexahedron using...
We study the approximation properties of some finite element subspaces of H(div;Ω) and H(curl;Ω) defined on hexahedral meshes in three dimensions. This work extends results previously obtained for quadrilateral H(div;Ω) finite elements and for quadrilateral scalar finite element spaces. The finite element spaces we consider are constructed starting from a given finite dimensional space of vector fields on the reference cube, which is then transformed to a space of vector fields on a hexahedron...
Low order edge elements are widely used for electromagnetic field problems. Higher order edge approximations are receiving increasing interest but their definition become rather complex. In this paper we propose a simple definition for Whitney edge elements of polynomial degree higher than one. We give a geometrical localization of all degrees of freedom over particular edges and provide a basis for these elements on simplicial meshes. As for Whitney edge elements of degree one, the basis is...
We propose transmission conditions of order 1, 2 and 3 approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses ε are at the order of the skin depth or essentially smaller. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t. to...
We propose transmission conditions of order 1, 2 and 3 approximating the shielding behaviour of thin conducting curved sheets for the magneto-quasistatic eddy current model in 2D. This model reduction applies to sheets whose thicknesses ε are at the order of the skin depth or essentially smaller. The sheet has itself not to be resolved, only its midline is represented by an interface. The computation is directly in one step with almost no additional cost. We prove the well-posedness w.r.t. to...
A nonlinear elliptic partial differential equation with homogeneous Dirichlet boundary conditions is examined. The problem describes for instance a stationary heat conduction in nonlinear inhomogeneous and anisotropic media. For finite elements of degree we prove the optimal rates of convergence in the -norm and in the -norm provided the true solution is sufficiently smooth. Considerations are restricted to domains with polyhedral boundaries. Numerical integration is not taken into account....