Weighted Bergman Projection on the Hartogs Triangle

We prove the $L^p$ regularity of the weighted Bergman projection on the Hartogs triangle, where the weights are powers of the distance to the singularity at the boundary. The restricted range of $p$ is proved to be sharp. By using a two-weight inequality on the upper half plane with Muckenhoupt weights, we can consider a slightly wider class of weights.Comment: The article has been revised. There are 23 pages in tota

Dense Matrices for Biofluids Applications

In this report, we focus on Biofluids problems, specifically the Stokes Equation. The method of regularized Stokeslets can be derived from bound- ary integral equations derived from the Lorentz reciprocal identity. When body forces are known, this is a direct numerical approximation of an in- tegral, resulting in a summation to determine the fluid velocity. In certain cases, which this report is focused on, we know the velocity and want to determine the forces on a structure immersed in a fluid. This results in a lin- ear system Af = u, where A is a square dense matrix. We study different methods to solve this system of equations to determine the force f on the structure. For solving a linear system with a dense coefficient matrix, the backslash command in MATLAB can be used. This will use an efficient and robust direct method for solving a smaller matrix, but this is not an efficient method for a large, dense coefficient matrix. For a large, dense coefficient ma- trix, we will explore other direct methods as well as several iterative methods to determine computation time and error on a test case with an exact solu- tion. For direct methods, we will study backslash, LU factorization and QR factorization methods. For iterative methods, we stuied Jacobi, Gauss-Seidel, SOR, GMRES, CG, CGS, BICGSTAB and Schulz CG methods for these bioflu- ids applications. All of these methods have different requirements. For our coefficient matrix A, we identified specific properties and then used proper methods, both direct and iterative. Result showed that iterative methods are more efficient then direct method for large size A. Schulz CG was slower but had a smaller error for the test case where there was an exact solution