Waldemar Rachowicz

Toward a universal h-p adaptive finite element strategy, part 1. Constrained approximation and data structure

A portable, power operated, hand cultivator comprising a frame having a motor supported thereon which, through a transmission, oscillates two or more generally vertically disposed cultivator tines extending downwardly from the frame. A handle is provided for easy control and manipulation of the device.

Toward a universal h-p adaptive finite element strategy, part 2. A posteriori error estimation

A harrow having two horizontal elongated tined members the ends of which are driven around substantially vertical axes and including a soil contacting elongated element mounted to each of the tined members.

Toward a universal h-p adaptive finite element strategy part 3. design of h-p meshes

Abstract In this third paper in our series, the issue of designing h-p meshes which are optimal in some sense is addressed. Criteria for h-p meshes are derived which are based on the idea of minimizing the estimated error over a mesh with a fixed number of degrees-of-freedom. An optimization algorithm is developed based on these criteria and is applied to several model one-dimensional and two-dimensional elliptic boundary-value problems. Numerical results indicate that the approach can lead to exponential rates-of-convergence.

A Fully Automatic hp -Adaptivity

We present an algorithm, and a 2D implementation for a fully automatic hp-adaptive strategy for elliptic problems. Given a mesh, the next, optimally refined mesh, is determined by maximizing the rate of decrease of the hp-interpolation error for a reference solution. Numerical results confirm optimal, exponential convergence rates predicted by the theory of hp methods.

De Rham diagram for hp finite element spaces

Abstract We prove that the hp finite elements for H (curl) spaces, introduced in [1], fit into a general de Rham diagram involving hp approximations. The corresponding interpolation operators generalize the notion of hp interpolation introduced in [2] and are different from the classical operators of Nedelec and Raviart-Thomas.

An hp-adaptive finite element method for electromagnetics: Part 1: Data structure and constrained approximation

Abstract This is the first of papers describing an implementation of the hp -adaptive, mixed Finite Element (FE) method for the solution of steady-state Maxwells equations proposed in L. Demkowicz and L. Vardapetyan (Modeling of electromagnetic absoption/scattering problems using hp -adaptive finite elements, Computer Methods in Applied Mechanics and Engineering, 152, 1998, 103–124). The discretization is defined on a hybrid grid consisting of both triangles and quads and allows for both h - and p -refinements of the mesh. The paper focuses on the data structure and constrained approximation issues and provides a number of illustrative examples.

A new finite element method for solving compressible Navier-Stokes equations based on an operator splitting method and h-p adaptivity

A new finite element method for solving compressible Navier-Stokes equations is proposed. The method is based on a version of Strangs operator splitting and an h-p adaptive finite element approximation in space. This paper contains the formulation of the method with a detailed discussion of boundary conditions, a sample adaptive strategy and numerical examples involving compressible viscous flow over a flat plate with Reynolds numbers Re = 1000 and Re = 10,000.

An anisotropic h-type mesh-refinement strategy

Abstract An anisotropic h-type mesh refinement strategy for finite element approximations is presented. The strategy consists of subsecting quadrilateral elements in one of two possible directions. The decision which elements should be subsected and in which direction is based on minimization of interpolation errors. The method is especially suitable for approximate solutions which in some regions display almost one-dimensional behavior, for instance solutions with boundary layers.

An h–p Taylor—Galerkin finite method for compressible Euler equations

Abstract An extension of the familiar Taylor-Galerkin method to arbitrary h - p spatial approximations is proposed. Boundary conditions are analyzed and a linear stability result for arbitrary meshes is given, showing the unconditional stability for the parameter of implicitness α ⩾ 0.5. The wedge and blunt body problems are solved with both linear, quadratic and cubic elements and h -adaptivity, showing the feasibility of higher orders of approximation for problems with shocks.

An anisotropic h-adaptive finite element method for compressible Navier-Stokes equations

Abstract Application of the finite element method with significantly stretched elements to solve compressible Navier-Stokes equations is presented. Stretched elements are introduced in an adaptive process as a result of selective directional subsecting of elements. An adaptive strategy of such anisotropic h -type refinements was developed. It is based on minimization of the interpolation errors, reduction of the residual error indicators and an adaptive recovery of the viscous fluxes with the possibility to control their accuracy. An iterative solver with the convergence characteristics independent of the aspect ratio of elements is presented. The solver is the GMRES algorithm with the domain decomposition Schwarz-type preconditioner. The proposed techniques are used to solve a benchmark problem, a hypersonic high Reynolds number flow with strong shock-boundary layer interaction.