Pointwise Computation in an Ill-Posed Spherical Pseudo-Differential Equation
-
Sergei V. Pereverzyev
and
Pavlo Tkachenko
Abstract
In the present paper, we consider the approximation of the solution of an ill-posed spherical pseudo-differential equation at a given point. While the methods for approximating the whole solution are well-studied in Hilbert spaces, such as the space of square-summable functions, the computation of values of the solution at given points is much less studied. This can be explained, in particular, by the fact that for square-summable functions the functional of pointwise evaluation is, in general, not well defined. To overcome this limitation we adjust the regularized least-squares method of An, Chen, Sloan and Womersley [Siam J. Numer. Anal. 50 (2012), no. 3, 1513–1534] by using a special a posteriori parameter choice rule. We also illustrate our theoretical findings by numerical results for the reconstruction of the solution at a given point.
Funding source: Austrian Fonds Zur Förderung der Wissenschaftlichen Forschung (FWF)
Award Identifier / Grant number: P25424
© 2015 by De Gruyter
Articles in the same Issue
- Frontmatter
- Hybridization of Mixed High-Order Methods on General Meshes and Application to the Stokes Equations
- Finite Element Analysis of an Exponentially Graded Mesh for Singularly Perturbed Problems
- A Reliable Residual Based A Posteriori Error Estimator for a Quadratic Finite Element Method for the Elliptic Obstacle Problem
- On the Theory of TE Waves Guided by a Lossy Three-Layer Structure with General Nonlinear Permittivity
- Analysis of Model Variance for Ensemble Based Turbulence Modeling
- On the Finite Volume Multigrid Method: Comparison of Intergrid Transfer Operators
- Approximation of Semilinear Fractional Cauchy Problem
- Pointwise Computation in an Ill-Posed Spherical Pseudo-Differential Equation
- On Conditioning of Constraints Arising from Variationally Consistent Discretization of Contact Problems and Duality Based Solvers
- The High Order Method with Discrete TBCs for Solving the Cauchy Problem for the 1D Schrödinger Equation
Articles in the same Issue
- Frontmatter
- Hybridization of Mixed High-Order Methods on General Meshes and Application to the Stokes Equations
- Finite Element Analysis of an Exponentially Graded Mesh for Singularly Perturbed Problems
- A Reliable Residual Based A Posteriori Error Estimator for a Quadratic Finite Element Method for the Elliptic Obstacle Problem
- On the Theory of TE Waves Guided by a Lossy Three-Layer Structure with General Nonlinear Permittivity
- Analysis of Model Variance for Ensemble Based Turbulence Modeling
- On the Finite Volume Multigrid Method: Comparison of Intergrid Transfer Operators
- Approximation of Semilinear Fractional Cauchy Problem
- Pointwise Computation in an Ill-Posed Spherical Pseudo-Differential Equation
- On Conditioning of Constraints Arising from Variationally Consistent Discretization of Contact Problems and Duality Based Solvers
- The High Order Method with Discrete TBCs for Solving the Cauchy Problem for the 1D Schrödinger Equation
