Leszek Demkowicz

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.

Modeling of electromagnetic absorption/scattering problems using hp-adaptive finite elements

A model problem for the steady-state form of Maxwells equations is considered. The problem is formulated in a weak form using a Lagrange multiplier, laying down a foundation for a general class of novel hp-adaptive FE approximations. A convergence proof for affine elements is presented. The proposed method is illustrated and verified with a series of 2D experiments including elements with curved boundaries and nonhomogeneous media.

On an h-type mesh-refinement strategy based on minimization of interpolation errors☆

Abstract An adaptive finite element method is proposed which involves an automatic mesh refinement in areas of the mesh where local errors are determined to exceed a pre-assigned limit. The estimation of local errors is based on interpolation error bounds and extraction formulas for highly accurate estimates of second derivatives. Applications to several two-dimensional model problems are discussed. The results indicate that the method can be very effective for both regular problems and problems with strong singularities.

Solution of 3D-Laplace and Helmholtz equations in exterior domains using hp-infinite elements

Abstract This work is devoted to a convergence study for infinite element discretizations for Laplace and Helmholtz equations in exterior domains. The proposed approximation applies to separable geometries only, combining an hp FE discretization on the boundary of the domain with a spectral-like representation (resulting from the separation of variables) in the ‘radial’ direction. The presentation includes a convergence proof for the Laplace equation and a stability analysis for the variational formulation of the Helmholtz equation in weighted Sobolev spaces. The theoretical investigations are verified and illustrated with numerical examples for the exterior spherical domain.

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.

Mixed Finite Elements, Compatibility Conditions, and Applications

Since the early 70’s, mixed finite elements have been the object of a wide and deep study by the mathematical and engineering communities. The fundamental role of this method for many application fields has been worldwide recognized and their use has been introduced in several commercial codes. An important feature of mixed finite elements is the interplay between theory and application. Discretization spaces for mixed schemes require suitable compatibilities, so that simple minded approximations generally do not work and the design of appropriate stabilizations gives rise to challenging mathematical problems.

An adaptive characteristic Petrov-Galerkin finite element method for convection-dominated linear and nonlinear parabolic problems in two space variables

Abstract This paper is a continuation of previous work of the authors [2]. An adaptive scheme for the analysis of time-dependent parabolic problems defined on two-dimensional or three-dimensional space domains is developed which is based on a Petrov-Galerkin method for spatial approximation and the method of characteristics for the temporal approximation. Numerical examples are discussed which illustrate the efficiency and effectiveness of the method.