Piecewise Quadratic Finite Element Research Articles

A Least-Squares Mixed Finite Element Method for Biot's Consolidation Problem in Porous Media

A least-squares mixed finite element method for the coupled problem of flow and deformation is presented and analyzed in this paper. For the analysis, we restrict ourselves to fully saturated conditions for the flow process and to a linearly elastic material law for the deformation process. This is known in the literature as Biot's consolidation problem. For simplicity, the analysis is presented for the problem in two space dimensions. Our least-squares approach is motivated by the fact that all process variables, i.e., fluid pressure and flux as well as displacement field and stress tensor, are approximated directly by suitable finite element spaces. Ellipticity of the corresponding variational formulation is proven for the stationary case as well as for the subproblems arising at each step of an implicit time discretization in the general time-dependent case. Standard H1 -conforming piecewise linear and quadratic finite elements are used for the fluid pressure and for (each component of) the displacement, respectively. For the flux and stress components, the H(div)-conforming Raviart--Thomas spaces (of lowest order) are used. Computational results are presented for some two-dimensional test problems.

Higher order dual Lagrange multiplier spaces�for mortar finite element discretizations

Domain decomposition techniques provide a powerful tool for the numerical approximation of partial differential equations. Here, we consider mortar techniques for quadratic finite elements. In particular, we focus on dual Lagrange multiplier spaces. These non-standard Lagrange multiplier spaces yield optimal discretization schemes and a locally supported basis for the associated constrained mortar spaces. As a result, standard efficient iterative solvers as multigrid methods can be easily adapted to the nonconforming situation. We construct locally supported and continuous dual basis functions for quadratic finite elements starting from the discontinuous quadratic dual basis functions for the Lagrange multiplier space. In particular, we compare different dual Lagrange multiplier spaces and piecewise linear and quadratic finite elements. The optimality of the associated mortar method is shown. Numerical results illustrate the performance of our approach.

On the quadratic finite element approximation to the obstacle problem

In this paper, we obtain the error bound \(O\left(h^{\frac 32 -\epsilon}\right) \) for any \( \epsilon > 0 \), for the piecewise quadratic finite element approximation to the obstacle problem, without the hypothesis that the free boundary has finite length (see [3]).

Small data oscillation implies the saturation assumption

The saturation assumption asserts that the best approximation error in \(H^1_0\) with piecewise quadratic finite elements is strictly smaller than that of piecewise linear finite elements. We establish a link between this assumption and the oscillation of \(f=-\Delta u\), and prove that small oscillation relative to the best error with piecewise linears implies the saturation assumption. We also show that this condition is necessary, and asymptotically valid provided \(f\in L^2\).

A nonconforming mixed finite element for second-order elliptic problems

In a recent work, Hiptmair [Mathematisches Institut, M9404, 1994] has constructed and analyzed a family of nonconforming mixed finite elements for second-order elliptic problems. However, his analysis does not work on the lowest order elements. In this article, we show that it is possible to construct a nonconforming mixed finite element for the lowest order case. We prove the convergence and give estimates of optimal order for this finite element. Our proof is based on the use of the properties of the so-called nonconforming bubble function to control the consistency terms introduced by the nonconforming approximation. We further establish an equivalence between this mixed finite element and the nonconforming piecewise quadratic finite element of Fortin and Soulie [J. Numer. Methods Eng., 19, 505–520, 1983]. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 445–457, 1997

Superconvergence of recovered gradients of piecewise quadratic finite element approximations. Part II: L∞‐error estimates

Superconvergence properties in the L2 norm are derived for the recovered gradients of piecewise quadratic finite element approximations on triangular partitions for two-dimensional elliptic problems and systems, including the case of linear elasticity. The analysis covers problems defined on polygonal domains, where the solutions have low regularity. The effects of numerical integration are treated.

Superconvergence of recovered gradients of piecewise quadratic finite element approximations. Part I: L2‐error estimates

AbstractSuperconvergence properties in the L2 norm are derived for the recovered gradients of piecewise quadratic finite element approximations on triangular partitions for two‐dimensional elliptic problems and systems, including the case of linear elasticity. The analysis covers problems defined on polygonal domains, where the solutions have low regularity. The effects of numerical integration are treated.

Nonconforming finite element methods for the equations of linear elasticity

In the adaptation of nonconforming finite element methods to the equations of elasticity with traction boundary conditions, the main difficulty in the analysis is to prove that an appropriate discrete version of Korn’s second inequality is valid. Such a result is shown to hold for nonconforming piecewise quadratic and cubic finite elements and to be false for nonconforming piecewise linears. Optimal-order error estimates, uniform for Poisson ratio ν ∈ [ 0 , 1 / 2 ) \nu \in [0,1/2) , are then derived for the corresponding P 2 {P_2} and P 3 {P_3} methods. This contrasts with the use of C 0 {C^0} finite elements, where there is a deterioration in the convergence rate as ν → 1 / 2 \nu \to 1/2 for piecewise polynomials of degree ≤ 3 \leq 3 . Modifications of the continuous methods and the nonconforming linear method which also give uniform optimal-order error estimates are discussed.

Approximation of the vector potential for viscous incompressible flow via the constant stress finite element

A convergence analysis for a piecewise quadratic finite element method to solve vector biharmonic problems inR 3 in primal variables is presented. The application to the equations of the vector potential for the flow of incompressible viscous fluids is focused. For simplicity, only the particular case of stationary stokesian flows is treated in detail, but it is showed that the method applies as well to more complex flows of such fluids.

A decomposition-dualization approach for solving constrained convex minimization problems with applications to discretized obstacle problems

In this paper, we shall be concerned with the solution of constrained convex minimization problems. The constrained convex minimization problems are proposed to be transformable into a convex-additively decomposed and almost separable form, e.g. by decomposition of the objective functional and the restrictions. Unconstrained dual problems are generated by using Fenchel-Rockafellar duality. This decomposition-dualization concept has the advantage that the conjugate functionals occuring in the derived dual problem are easily computable. Moreover, the minimum point of the primal constrained convex minimization problem can be obtained from any maximum point of the corresponding dual unconstrained concave problem via explicit return-formulas. In quadratic programming the decomposition-dualization approach considered here becomes applicable if the quadratic part of the objective functional is generated byH-matrices. Numerical tests for solving obstacle problems in ?1 discretized by using piecewise quadratic finite elements and in ?2 by using the five-point difference approximation are presented.

A non‐conforming piecewise quadratic finite element on triangles

AbstractIt is shown in this paper that non‐conforming finite elements on the triangle using second‐degree polynomials can be easily built and used. Indeed they appear as an ‘enriched’ version of the standard piecewise quadratic six‐node element. This work is divided into two parts. In the first we present the basic properties of the element, namely how it can be built and basic error estimates. We also show that this element exhibits a very peculiar regularity property. In the second part we apply our element to the approximation of viscous incompressible flows and more generally to the approximation of incompressible materials.

Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation

Estimation of Linear Functionals on Sobolev Spaces with Application to Fourier Transforms and Spline Interpolation