A weighted least squares finite element method for elliptic problems with degenerate and singular coefficients

Abstract

We consider second order elliptic partial differential equations with coefficients that are singular or degenerate at an interior point of the domain. This paper presents formulation and analysis of a novel weighted-norm least squares finite element method for this class of problems. We propose a weighting scheme that eliminates the pollution effect and recovers optimal convergence rates. Theoretical results are carried out in appropriately weighted Sobolev spaces and include ellipticity bounds on the weighted homogeneous least squares functional, regularity bounds on the elliptic operator, and error estimates. Numerical experiments confirm the predicted error bounds.

Analysis of Schrödinger operators with inverse square potentials II: FEM and approximation of eigenfunctions in the periodic case

“…We have to mention Kato's ground breaking papers [30], where the self-adjointness of Schrödinger type Hamiltonians was proved and [31], where boundedness properties of the eigenfunctions and eigenvalues of these Hamiltonian operators was proved. Moreover, see [5,8,11,12,42,48,49] for other papers studying Hamiltonians with inverse square potentials, both from the point of view of physical and numerical applications. See also [9,10,13,14,15,16,20,23,46,47,50] for some related results.…”

“…We have to mention Kato's ground breaking papers [30], where the self-adjointness of Schrödinger type Hamiltonians was proved and [31], where boundedness properties of the eigenfunctions and eigenvalues of these Hamiltonian operators was proved. Moreover, see [5,8,11,12,42,48,49] for Date: September 23, 2018. V.N.…”

“…We think of stretching out the holes where the singularities of V are and compactifying the result using boundary spheres. It would be possible to carry out analysis similar to the calculations in this paper with only the assumption that ρ 2 V be smooth on T S. To simplify some calculations and to obtain closed form results, however, we will further require the resulting function Z to be constant on the blow up of each point in S, which can be reformulated as saying that ρ 2 V is also continuous on T. Our first assumption on the potential V is therefore (8) Assumption 1 :…”

A incremental localization based on feasible weighted least squares

“…It has been proved that when the matrix W is the reciprocal of the variance matrix of the range error under the condition that the ratio of measurement error to the distance is independent Gaussian random variables, the error variance by OWLS is minimal [15,16]. But in reality environment, the variance of the error term is unknown; therefore, if OWLS is solved, the weight matrix W needs to be taken according to the actual situation.…”

Incremental Localization Algorithm Based on Multivariate Analysis

“…Through Schwarz inequality [17, 18], it is proved that when the matrix Ω is the reciprocal of the variance matrix of the range error under the condition that the ratio of range error to the distance is independent Gaussian random variables, the error variance by WLS is minimal. But in reality, the variance of the error term is unknown; therefore, if WLS is solved, the weight needs to be taken according to the actual situation.…”