Müntz-Galerkin Methods and Applications to Mixed Dirichlet-Neumann Boundary Value Problems
نویسندگان
چکیده
Solutions for many problems of interest exhibit singular behaviors at domain corners or points where boundary condition changes type. For this type of problems, direct spectral methods with usual polynomial basis functions do not lead to a satisfactory convergence rate. We develop in this paper a Müntz-Galerkin method which is based on specially tuned Müntz polynomials to deal with the singular behaviors of the underlying problems. By exploring the relations between Jacobi polynomials and Müntz polynomials, we develop efficient implementation procedures for the MüntzGalerkin method, and provide optimal error estimates. As examples of applications, we consider the Poisson equation with mixed Dirichlet-Neumann boundary conditions, whose solution behaves like O(r1/2) near the singular point, and demonstrate that the Müntz-Galerkin method greatly improves the rates of convergence of the usual spectral method.
منابع مشابه
Challenges and Applications of Boundary Element Domain Decomposition Methods
Boundary integral equation methods are well suited to represent the Dirichlet to Neumann maps which are required in the formulation of domain decomposition methods. Based on the symmetric representation of the local Steklov– Poincaré operators by a symmetric Galerkin boundary element method, we describe a stabilized variational formulation for the local Dirichlet to Neumann map. By a strong cou...
متن کاملHp -version Discontinuous Galerkin Finite Element Methods for Semilinear Parabolic Problems
We consider the hp–version interior penalty discontinuous Galerkin finite element method (hp–DGFEM) for semilinear parabolic equations with mixed Dirichlet and Neumann boundary conditions. Our main concern is the error analysis of the hp–DGFEM on shape–regular spatial meshes. We derive error bounds under various hypotheses on the regularity of the solution, for both the symmetric and non–symmet...
متن کاملAnalysis of hypersingular residual error estimates in boundary element methods for potential problems
A novel iteration scheme, using boundary integral equations, is developed for error estimation in the boundary element method. The iteration scheme consists of using the boundary integral equation for solving the boundary value problem and iterating this solution with the hypersingular boundary integral equation to obtain a new solution. The hypersingular residual r is consistently defined as t...
متن کاملNUMERICAL SOLUTIONS OF SECOND ORDER BOUNDARY VALUE PROBLEM BY USING HYPERBOLIC UNIFORM B-SPLINES OF ORDER 4
In this paper, using the hyperbolic uniform spline of order 4 we develop the classes of methods for the numerical solution of second order boundary value problems (2VBP) with Dirichlet, Neumann and Cauchy types boundary conditions. The second derivativeis approximated by the three-point central difference scheme. The approximate results, obtained by the proposed method, confirm theconvergence o...
متن کاملExponential Convergence of the H ? P Version Bem for Mixed Bvp's on Polyhedrons
We analyze the h-p version of the bem for mixed Dirichlet Neumann problems of the Laplacian in polyhedral domains. Based on a regularity analysis of the solution in count-ably normed spaces we show that the boundary element Galerkin solution of the h-p version converges exponentially fast on geometrically graded meshes.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- SIAM J. Scientific Computing
دوره 38 شماره
صفحات -
تاریخ انتشار 2016