Abstract: In this paper, we present a wrapper around MUMPS solver, called Hierarchical Solver Wrapper (HSW) that allows reducing its memory usage and execution time. The wrapper is dedicated for domain decomposition based parallel finite element method computations on Linux cluster. It utilizes the identical interface as parallel MUMPS with entries provided as lists of non-zero entries distributed among multiple processors. It wraps around MUMPS solver, performing two level hierarchical calls. First, it calls multiple sequential MUMPS solvers, one per each processor. They compute Schur complements of the interior nodes on the interface nodes over each subdomain. It deallocates the sequential MUMPS solvers on every processor to minimize the memory usage. It keeps only original lists of non-zero entries as well as lists of non-zero entries obtained from the computed Schur complement. It submits all the Schur complements to one parallel MUMPS solver and asks for the skeleton problem solution. Later, it restores the upper part of the local matrix to perform local backward substitutions with Schur complements replaced by the identity matrices and right-hand-sides superseded by the local part of the global solution. The wrapper is tested with latest version of MUMPS over the three-dimensional isogeometric analysis computations. It enables for reduction of the memory usage and also the execution time, in comparison with a single parallel MUMPS call with distributed lists of non-zero entries.

Abstract: In context of adaptive finite element methods, we explore the possibility of employing lowest-order non-fitting meshes for solving Maxwell´s equations. In a fitting mesh, the physical interfaces of the propagation medium must be aligned with cell faces. On the other hand, this constraint is removed for non-fitting meshes, so that a larger choice of cells is possible. As a result, non-fitting meshes can simplify the implementation and/or reduce the computational cost of the associated finite-element method. Unfortunately, non-fitting meshes can also cause an accuracy loss. Indeed, the solution of Maxwell's equations can become singular inside mesh cells due to material discontinuities. In this work, we carefully analyze the accuracy loss due to the use of non-fitting meshes. We derive error estimates that are further confirmed via numerical experiments. The main conclusion is that electric and magnetic fields exhibit different convergence rates in the L^2-norm for the case of non-fitting meshes. In particular, if the user is interested in the magnetic field, non-fitting meshes produce solutions of similar quality to those obtained using fitting meshes.

Abstract: In this talk, we focus on the application of the isogeometric analysis for tumor growth simulations. We present an application of the fast algorithm for isogeometric L2 projections for simulations of the tumor growth. We utilize the of PDE describing the model of the melanoma growth, including tumor cell density, flux, pressure, extracellular and degraded extracellular matrices, previously used in finite difference simulations. We also use a discrete model of the vasculature that provides an oxygen source to the system. The system is solved using explicit scheme and fast isogeometric L2 projections, utilizing the alternating directions solver. Every 10-time steps of the simulation we couple our continuous model with the discrete vasculature model. The presentation is concluded with the two-dimensional numerical results. We show that explicit dynamics simulation needs around 30,000 times steps of the alternating direction solver to be performed in two-dimensions.

Abstract: This work proposes the use of an alternative error representation for Goal-Oriented Adaptivity (GOA) in context of steady state convection dominated diffusion problems. It introduces an arbitrary operator for the computation of the error of an alternative adjoint problem. From the new representation, we derive element-wise estimators to drive the adaptive algorithm. The method is applied to a one dimensional (1D) steady state convection dominated diffusion problem with homogeneous Dirichlet boundary conditions. This problem exhibits a boundary layer that produces a loss of numerical stability. The new error representation delivers sharper error bounds. When applied to a p-GOA Finite Element Method (FEM), the alternative error representation captures earlier the boundary layer, despite the existing spurious numerical oscillations.

Abstract: We propose a way of performing isogeometric finite element method (IGA-FEM) computations over h refined grids with B-spline basis functions. Namely, we propose to use the B-spline basis functions defined over patches of elements with C0 separators between the refinement levels. Our definition of the B-splines and C0 separators allows introduction of arbitrary order B-splines over 2D grids refined towards singularities. We also present an algorithm for construction of element partition trees (EPT) over h refined grids with modified B-splines. The EPT allows generating an ordering which gives a linear computational cost of the multi-frontal solver over 2D grids refined towards a point or an edge singularity. We present the algorithm for transforming the EPT into an ordering. We also verify the linear computational cost of the proposed method on grid with point and edge singularity. We compare our method to h-adaptive finite element method (h-FEM) computations with Lagrange polynomials.