Special Finite Element Methods for a Class of Second Order Elliptic Problems with Rough Coefficients (original) (raw)

Remarks on mixed finite element methods for problems with rough coefficients

Mathematics of Computation, 1994

This paper considers the finite element approximation of elliptic boundary value problems in divergence form with rough coefficients. The solution of such problems will, in general, be rough, and it is well known that the usual (Ritz or displacement) finite element method will be inaccurate in general. The purpose of the paper is to help clarify the issue of whether the use of mixed variational principles leads to finite element schemes, i.e., to mixed methods, that are more accurate than the Ritz or displacement method for such problems. For one-dimensional problems, it is well known that certain mixed methods are more accurate and robust than the Ritz method for problems with rough coefficients. Our results for two-dimensional problems are mostly of a negative character. Through an examination of examples, we show that certain standard mixed methods fail to provide accurate approximations for problems with rough coefficients except in some special situations.

Higher order approximations to the boundary conditions for the finite element method

Mathematics of Computation, 1976

We consider here the approximation of essential boundary conditions for the finite element solutions of second order elliptic equations in two dimensions. Nonhomogeneous boundary conditions on curved boundaries are treated. The approach is to use trial functions which interpolate (in a generalized sense) functions satisfying the boundary conditions. The work is directed to showing in what manner this interpolation should be done to achieve the maximum accuracy and computational simplicity. These methods can be used to construct approximations of arbitrary high order of accuracy. Several examples are given.

Mixed finite element methods for nonlinear elliptic problems: Thep-version

Numer Method Partial Differ E, 1996

Mixed finite element methods for strongly nonlinear second order elliptic problems are proposed and analyzed. Existence and uniqueness of the approximate solution are demonstrated using a fixed point argument. Convergence and stability of the method are proved both with respect to mesh refinement and increase of the degree of the approximating polynomials. The analysis is carried out in detail using Raviart-Thomas-Nedelec spaces as an example. Numerical results for minimal surface problems are obtained using Brezzi-Douglas-Marini spaces. Graphs of the approximate solutions are presented for two different problems.

A Systematic Study on Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

Journal of Scientific Computing, 2017

This article provides a systematic study for the weak Galerkin (WG) finite element method for second order elliptic problems by exploring polynomial approximations with various degrees for each local element. A typical local WG element is of the form P k (T) × P j (∂ T) P (T) 2 , where k ≥ 1 is the degree of polynomials in the interior of the element T , j ≥ 0 is the degree of polynomials on the boundary of T , and ≥ 0 is the degree of polynomials employed in the computation of weak gradients or weak first order partial derivatives. A general framework of stability and error estimate is developed for the corresponding numerical solutions. Numerical results are presented to confirm the theoretical results. The work reveals some previously undiscovered strengths of the WG method for second order elliptic problems, and the results are expected to be generalizable to other type of partial differential equations.

On Solving Nonlinear Elasticity Problems Using a Boundary-Elements-Based Solution Method

Method. Appl. Mech. 2023, 4, 0–17., 2023

Abstract: The attractiveness of the boundary element method—the reduction in the problem dimension by one—is lost when solving nonlinear solid mechanics problems. The point collocation method applied to strong-form differential equations is appealing because it is easy to implement. The method becomes inaccurate in the presence of traction boundary conditions, which are inevitable in solid mechanics. A judicious combination of the point collocation and the boundary integral formulation of Navier’s equation allows a pure boundary element method to be obtained for the solution of nonlinear elasticity problems. The potential of the approach is investigated in some simple examples considering isotropic and anisotropic material models in the total Lagrangian framework. Keywords: elasticity; anisotropy; large deformation; BEM; meshfree strong form; radial basis function

A Finite Element Method for Nonlinear Elliptic Problems

SIAM Journal on Scientific Computing, 2013

We present a continuous finite element method for fully nonlinear elliptic equations. The tools we use are (1) a Newton linearisation, yielding a sequence of linear PDEs in nonvariational form and (2) the discretisation proposed in [LP11] allowing us to work directly on the strong form of a linear PDE. An added benefit to making use of this discretisation method is that a recovered (finite element) Hessian is a biproduct of the solution process. Benchmark numerical results illustrate the convergence properties of the scheme for some test problems.

On mixed finite element techniques for elliptic problems

International Journal of Mathematics and Mathematical Sciences, 1983

The main aim of this paper is to consider the numerical approximation of mildly nonlinear elliptic problems by means of finite element methods of mixed type. The technique is based on an extended variational principle, in which the constraint of interelement continuity has been removed at the expense of introducing a Lagrange multiplier.It is shown that the saddle point, which minimizes the energy functional over the product space, is characterized by the variational equations. The eauivalence is used in deriving the error estimates for the finite element approximations. We give an example of a mildly nonlinear elliptic problem and show how the error estimates can be obtained from the general results.

A mixed finite element method for a strongly nonlinear second-order elliptic problem

Mathematics of Computation, 1995

The approximation of the solution of the first boundary value problem for a strongly nonlinear second-order elliptic problem in divergence form by the mixed finite element method is considered. Existence and uniqueness of the approximation are proved and optimal error estimates in L2 are established for both the scalar and vector functions approximated by the method.

Mixed finite element methods for nonlinear elliptic problems: the h-p version

Journal of Computational and Applied Mathematics, 1997

Special Finite Element Methods for a Class of Second Order Elliptic Problems with Rough Coefficients (original) (raw)

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