In the present paper we study the approximation of functions with bounded mixed derivatives by sparse tensor product polynomials in positive order tensor product Sobolev spaces. We introduce a new sparse polynomial approximation operator which exhibits optimal convergence properties in L and tensorized H 0 simultaneously on a standard k-dimensional cube. In the special case k = 2 the suggested ...

Thermo-hyperelastic problems at finite strains belong to a category of non-linear coupled problems that impose challenges on their numerical treatment. We present the weak-form for a 1-D coupled, stationary, thermo-hyperelastic system with constant or temperature-dependent material properties. The coupled system is discretized by a ‘monolithic’ high-order finite element method (p-FEM) based on ...

In this note we consider the continuous Galerkin time-stepping method of arbitrary order as a possible discretization scheme nonlinear initial value problems. addition, develop and generalize well-known existing result for discrete solution by applying general linearizing procedure to including also simplified Newton procedure. particular, presented existence results are implied choosing suffic...

In this paper, the second order boundary value problem −∇·(A(x, y)∇u) = f is discretized by the Finite Element Method using piecewise polynomial functions of degree p on a triangular mesh. On the reference element, we define integrated Jacobi polynomials as interior ansatz functions. If A is a constant function on each triangle and each triangle has straight edges, we prove that the element sti...

We present the mathematical analysis for the convergence of an h version Finite Element Method (FEM) with piecewise polynomials of degree p ≥ 1, de ned on an exponentially graded mesh. The analysis is presented for a singularly perturbed reaction-di usion and a convection-di usion equation in one dimension. We prove convergence of optimal order and independent of the singular perturbation param...

The performance of the finite element method (FEM) may be improved in three ways. The first is by the h-version to refine the finite element mesh and the second is by the p-version to increase the order of polynomial shape functions for a fixed mesh. The third is simultaneously to refine the mesh and increase the degrees of elements uniformly or selectively, called the h-p version, which is a c...

The hp version of the finite element method (hp-FEM) combined with adaptive mesh refinement is a particularly efficient method for solving partial differential equations (PDEs) because it can achieve an exponential convergence rate in the number of degrees of freedom. hp-FEM allows for refinement in both the element size, h, and the polynomial degree, p. Like adaptive refinement for the h versi...