Adaptive galerkin method with relevant basis functions for pdes with boundary conditions

As a useful tool for solving partial differential equations, Galerkin method has been developed for solving different problems through constructing different types of basis functions. Previous construction methods mainly focused on constructing common and optimal basis functions, neglecting the effect of the known information existing in differential equation itself. To adequately utilize the existing information, relevant basis function (RBF) based on optimal thought is proposed in this paper. The concept of relevant basis function is defined, and its properties, including similarity, adaptability and optimality, are described. Different from traditional basis functions, RBFs are formed by two parts. Ones are the known relevant basis functions, constructed by utilizing the known conditions reflected by the form of differential equation, and the others are the unknown relevant basis functions with the known form determined by the known conditions, including undetermined part. After constructing relevant basis functions, the adaptive Galerkin method with relevant basis functions is designed for solving partial differential equations with boundary conditions, mainly including two aspects. One is that the coefficients of basis functions are obtained by Galerkin method, and the other is that the undetermined part of unknown relevant basis functions is solved adaptively by iterative method. Numerical examples demonstrate that the adaptive Galerkin method with relevant basis functions is flexible and accurate with economical algorithm for solving partial differential equations with boundary conditions.