Abstract From the literature it is known that conjugate gradient method with domain decomposition preconditioners one of most efficient methods for solving systems linear algebraic equations resulting from p ‐version finite element discretizations elliptic boundary value problems. One ingredient such a preconditioner related to Dirichlet In case Poisson's equation, we present problems which can be interpreted as stiffness matrix K h,k h discretization special degenerated problem. We...

From the literature it is known that conjugate gradient method with domain decomposition preconditioners one of most efficient methods for solving systems linear algebraic equations resulting from p-version finite element discretizations elliptic boundary value problems. The ingredients such a preconditioner are Schur complement, related to Dirichlet problems in subdomains, and an extension operator boundaries subdomains into their interior. In case Poisson's equation, we propose which can...

In this work, we develop a posteriori error control for generalized Boussinesq model in which thermal conductivity and viscosity are temperature-dependent. Therein, the stationary Navier–Stokes equations coupled with heat equation. The problem is modeled solved monolithic fashion. focus on multigoal-oriented estimation dual-weighted residual method an adjoint utilized to obtain sensitivity measures respect several goal functionals. localization achieved help of partition-of-unity weak...

H(curl) conforming finite element discretizations are a powerful tool for the numerical solution of system Maxwell's equations in electrodynamics. In this paper we construct basis high-order function space 3 dimensions. We introduce set hierarchic functions on tetrahedra with property that both L2-inner product and H(curl)-inner sparse respect to polynomial degree. The construction relies tensor-product based structure properly weighted Jacobi polynomials as well an explicit splitting into...

This paper is devoted to the fast solution of interface concentrated finite element equations. The schemes are constructed on basis a nonoverlapping domain decomposition where conforming boundary approximation used in every subdomain. Similar methods, total number unknowns per subdomain behaves like $O((H/h)^{(d-1)})$, H, h, and d denote usual scaling parameter subdomains, average discretization boundaries, spatial dimension, respectively. We propose analyze primal dual substructuring...

Abstract From the literature it is known that orthogonal polynomials as Jacobi can be expressed by hypergeometric series. In this paper, authors derive several contiguous relations for terminating multivariate With these one prove recursion formulas of those This theoretical result allows to compute integrals over products in a very efficient recursive way. Moreover, present an application numerical analysis where used algorithms which approximate solution boundary value problem partial...

.From the literature, it is known that choice of basis functions in hp-FEM heavily influences computational cost order to obtain an approximate solution. Depending on reference element, suitable tensor product like Jacobi polynomials with different weights lead optimal properties due condition number and sparsity. This paper presents biorthogonal primal mentioned above. The authors investigate hypercubes simplices as elements, well cases H1 H(Curl). can be expressed sums products maximal two...

Abstract In this contribution, we apply adaptive finite elements to the Boussinesq model. Adaptivity is achived with goal‐oriented error control and local mesh refinement. The principle goal motivated from laser material processing waveguide writing in which starts flow due laser‐induced heat generation. Flow of decribed by Boussinseq equations. Our model substantiated some numerical tests order show capacities our schemes.

Abstract In this paper we consider higher order shape functions for finite elements on a triangle. On the reference element Dubiner‐like ansatz based suitable integrated Jacobi polynomials are chosen. It can be proved that corresponding mass and stiffness matrices sparse all polynomial degree p . Due to orthogonal relations between exact values of entries matrix determined. Using symbolic computation, find simple recurrence which allow us compute remaining nonzero in optimal arithmetic complexity.