Quasi-optimal Error Estimates Research Articles

Boundary integral equations for magnetic screens in ℝ3

SynopsisA boundary integral method is developed for the scattering of electromagnetic waves at thin obstacles. The exterior boundary value problem for the vector Helmholtz equation with given Neumann data on an open surface piece (screen S) is converted into a system of integral equations for the jumps of the tangential component of the field and its divergence across the screen. A slight modification of the Cauchy data yields a strongly elliptic system of pseudodifferential equations on S which can therefore be used for numerical computations using Galerkin's procedure. The resulting boundary integral equations are analysed using pseudodifferential operator calculus. The principal symbol concept, together with the Wiener–Hopf technique, are used to derive existence and regularity results for the solutions to the boundary integral equations. Quasi-optimal error estimates in the energy norm are given for the numerical scheme.

A direct boundary integral equation method for transmission problems

A system of integral equations for the field and its normal derivative on the boundary in acoustic or potential scattering by a penetrable homogeneous object in arbitrary dimensions is presented. The system contains the operators of the single and double layer potentials, of the normal derivative of the single layer, and of the normal derivative of the double layer potential. It defines a strongly elliptic system of pseudodifferential operators. It is shown by the method of Mellin transformation that a corresponding property, namely a Gårding's inequality in the energy norm, holds also in the case of a polygonal boundary of a plane domain. This yields asymptotic quasioptimal error estimates in Sobolev spaces for the corresponding Galerkin approximation using finite elements on the boundary only.

Numerical solution of the obstacle problem by the penalty method

In a previous paper we considered the possibility of using the penalty method to get approximations of the solution of elliptic variational inequalities. In this note we show that the same idea can be used for the treatment of parabolic problems. By coupling the penalty parameter e and the discretization parametersh and δt quasi-optimal error estimates are derived in suitable norms.

Boundary element method for membrane and torsion crack problems

Abstract Here we present as an application of [12] an improved Galerkin method for the boundary integral equations governing a plane interface problem. Membrane and torsion crack problems can be treated by slight modifications. A tedious analysis incorporating the Mellin transform shows that the coupled system of integral equations—with some Fredholm equations of the second kind and some of the first kind—on the boundary curve Γ is strongly elliptic, i.e., there holds a Garding inequality. This property implies convergence of almost optimal order of the Galerkin procedure. The use of singularity functions together with regular finite elements on Γ provides convergence results in a scale of Sobolev spaces and even quasi-optimal asymptotic error estimates for the stress intensity factors. These factors are computed directly by our Galerkin scheme, i.e., no additional computations are needed. The Galerkin method is implemented by an appropriate numerical integration leading to a Galerkin collocation. The latter is a modified collocation method which can be easily implemented on computing machines.

The normal dervative of the double layer potential on polygons and galerkin approximation

The method of Mellin transformation is applied to the normal derivative of the double layer potential on a plane polygon. For the corresponding boundary integral ope¬rator continuity, a Carding inequality, and regularity of the solution are proved in Sobolev spaces. The results are applied to show quasi-optimal asymptotic error estimates for the Galerkin approximation procedure which uses the explicit de¬composition of the solution into singular functions at the corners and a smooth remainder

A New Approximation Method for the Helmholtz Equation in an Exterior Domain

In this paper we consider the numerical solution of the Helmholtz equation in an exterior domain in $\mathbb{R}^3 $. We introduce a new variational formulation of the problem involving a nonlocal boundary condition. We show that the discrete bilinear form associated with our variational formulation satisfies the “inf-sup” condition, and we give a quasi-optimal error estimate for the approximate solution. Moreover, we show a certain density property of our subspaces which plays a crucial role in the justification of the method.

Error estimates for mixed methods

Abstract : This paper presents abstract error estimates for mixed methods for the approximate solution of elliptic boundary value problems. These estimates are then applied to obtain quasi-optimal error estimates in the usual Sobolev norms for four examples: three mixed methods for the biharmonic problem and a mixed method for 2nd order elliptic problems. (Author)

Convergence of an equilibrium finite element model for plane elastostatics

An equilibrium triangular block-element, proposed by Watwood and Hartz, is subjected to an analysis and its approximability property is proved. If the solution is regular enough, a quasi-optimal error estimate follows for the dual approximation to the mixed boundary value problem of elasticity (based on Castigliano's principle). The convergence is proved even in a general case, when the solution is not regular.

Error estimates for Galerkin methods for quasilinear parabolic and elliptic differential equations in divergence form

Conditions are given for the validity of long range quasi optimal error estimates of the Galerkin method for quasilinear parabolic equations. If the corresponding nonlinear elliptic operator satisfies a coercivity condition, the stationary problem may be approximated by the numerical solution of the parabolic problem when the time variablet is large enough, starting with an arbitrary initial function. Some practical applications are discussed.