Error analysis of an implicit Galerkin meshfree scheme for general second-order parabolic problems

Abstract

An implicit Galerkin meshfree scheme based on the element-free Galerkin (EFG) and the backward Euler methods is analyzed for general second-order parabolic problems. The penalty method is used to impose Dirichlet boundary conditions, and a priori approximation error has been derived to confirm its validity. The existence and uniqueness of the weak solution for the penalized parabolic problems under homogeneous mixed boundary conditions are proved, offering an insight for numerical discretizations. With an orthogonal projection being defined, the continuous error bounds with the penalty factor for the semi-discrete approximation are obtained, and meanwhile, the discrete error bounds with the penalty factor for the fully discrete approximation are also achieved, which provide a theoretical explanation for the value of a reasonable penalty factor. Finally, numerical examples are conducted to verify the theoretical results.