Numerical Methods And Convergence Analysis Of Boundary Value And Inverse Boundary Value Problems For Elliptic Equations

Boundary value problems for elliptic equations describe many steady state problems,like equilibrium problems in elastic mechanics,electronic density of conductor.Due to the complexity of problem domains and boundary conditions,it is difficulty to solve this kind of boundary value problems exactly.Therefore,it has great significance to find numerical methods to solve this kind problems and to prove the convergence theorems of numerical solutions to exact ones.Inverse boundary value problems for elliptic equations appear in many practical applications,like acoustic scattering field,tomography,and nondestructive testing.This kind of problems is ill-posed in the sense that small measurement errors can induce large errors in computation.Therefore,stable reconstruction methods and convergence analysis for numerical methods are important for inverse problems in real life.Our work are focus on construct numerical schemes and convergence analysis for numerical methods of these two kinds problems.For boundary value problems of elliptic equations,we use a nodal based smoothed point interpolation method(NS-RPIM).Shape functions are built by using the radial point interpolation method with pure radial basis functions.The smoothed Galerkin weak form is applied to construct discretized system equations.Triangle background cells with Voronio smoothing domains and quadrilateral background cells with equally-shared areas are used.For these two kinds smoothing domains,we use different node selection strategies.It is found that in all cases that the NS-RPIM provides results with higher convergence rates for distorted mesh compared with results by the FEM.In addition,in all numerical examples,solutions for the energy norm by NS-RPIM are found to be upper bounds with respect to the FEM counterparts and even to exact solutions.For inverse boundary value problems of elliptic equations,by applied overdetermined Kansa method,we propose two numerical schemes and prove their convergence results.By imposing pointwise equality constraint conditions to control errors on accessible boundary,we proposed an adaptive least squares meshfree method.The method uses at most three steps in computing.The convergence of numerical solution to the exact one is proved based on the scattered data approximation theory in reproducing kernel Hilbert space.Tikhonov regularization term naturally appears in least squares formulations and regularization parameter can be fixed in convergence proof.By imposing quadratic inequality conditions to decrease computing errors on accessible boundary,we propose a optimization reconstruction scheme.We formulate semidescretized solutions as solutions of optimization problems with quadratic inequality conditions.Based on the theory of reproducing kernel Hilbert space,convergence of the semi-descretized solution is proved.By using collocation point sets,we define the fully discretized solution.By fractional sampling inequalities,convergence of the fully discretized solution is proved.Numerical experiments in both two dimension and three dimension show the effectiveness of above two proposed numerical algorithms.Numerical results show that our schemes can reconstruct stable and high accuracy solutions in estimating the unknown data from the noisy Cauchy data,especially in the case when noise levels are large.