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A finite element method for a monolithic quasi-Lagrangian formulation of a fluid–porous structure interaction problem with a corrected balance of stresses on the fluid–structure interface is considered. Deformations of the elastic medium are not necessarily small and are modelled using Saint Venant–Kirchhoff (SVK) constitutive relation. The stability of the method is proved in a form of energy bound for the finite element solution.

Palliation of congenital single ventricle heart defects suggests multi-stage surgical interventions that divert blood flow from the inferior and superior vena cava directly to the right and left pulmonary arteries, skipping the right ventricle. Such system with cavopulmonary anastomoses and single left ventricle is called Fontan circulation, and the region of reconnection is called the total cavopulmonary connection (TCPC). Computational blood flow models allow clinicians to predict the results of the Fontan operation, to choose an optimal configuration of TCPC and thus to reduce negative postoperative consequences. We propose a two-scale (1D3D) haemodynamic model of systemic circulation for a patient who has underwent Fontan surgical operation. We use CT and 4D flow MRI data to personalize the model. The model is tuned to patient’s data and is able to represent measured time-averaged flow rates at the inlets and outlets of TCPC, as well as pressure in TCPC for the patient in horizontal position.We demonstrate that changing to quiescent standing position leads to other patterns of blood flow in regional (TCPC) and global haemodynamics. This confirms clinical data on exercise intolerance of Fontan patients.

This work presents a new approach to modelling of free surface non-Newtonian (viscoplastic or viscoelastic) fluid flows on dynamically adapted octree grids. The numerical model is based on the implicit formulation and the staggered location of governing variables. We verify our model by comparing simulations with experimental and numerical results known from the literature.

The paper discusses a stabilization of a finite element method for the equations of fluid motion in a time-dependent domain. After experimental convergence analysis, the method is applied to simulate a blood flow in the right ventricle of a post-surgery patient with the transposition of the great arteries disorder. The flow domain is reconstructed from a sequence of 4D CT images. The corresponding segmentation and triangulation algorithms are also addressed in brief.

In this work we analyze the impact of left ventricular assist devices on the systemic circulation in subjects with heart failure associated with left ventricular dilated cardiomyopathy. We use an integrated model of the left heart and blood flow in the systemic arteries with a left ventricular assist device. We study the impact of the rotation speed of the pump on haemodynamic characteristics of distal arteries. We identify the rotation speed for simultaneous recovery of the healthy average values in all systemic arteries, the heart and the aorta. Our numerical experiments show that blood distribution over the graph of systemic vessels does not depend on flow regimes in ascending aorta. We also observe that the optimal pump rotation speed changes in the atherosclerotic vascular network and depends on stenoses localization.

Human knee is one of the most complex joints. Different reasons may lead to knee instability. A personalized mathematical model of the knee may improve both diagnostic procedure and knee surgery outcomes. Such models require accurate geometric representation of bones and attachment sites of ligaments and tendons. This paper addresses automatic segmentation of knee bones and detection of origins and insertions for tendons and ligaments. The approach is based on anatomical features of bones and landmarks of tendons/ligaments attachments on the CT images. It provides a tool for the design of patient-specific geometrical knee models.

Aortic valve disease accounts for 45% of deaths from heart valve diseases.% \cite{Coffey2015}. An appealing approach to treat aortic valve disease is surgical replacement of the valve leaflets based on chemically treated autologous pericardium. This procedure is attractive due to its low cost and high effectiveness. We aim to develop a computational technology for patient-specific assessment of reconstructed aortic valve function that can be used by surgeons at the preoperative stage. The framework includes automatic computer tomography image segmentation, mesh generation, simulation of valve leaflet deformation. The final decision will be based on uncertainty analysis and leaflet shape optimization. This paper gives a proof of concept of our methodology: simulation methods are presented and studied numerically.

The article introduces a finite-volume method for the Darcy problem in heterogeneous anisotropic media. The method is based on the mixed formulation for the pressure and its gradient. The method is stable despite collocation of both pressure and its gradient at cell centers and demonstrates the first order convergence on numerous benchmarks as well as good monotonicity property. The method produces quasi-definite matrix, which is numerically shown to have good asymptotics of the condition number. Our flux discretization method is a realization of our more general concept of stable flux discretization for saddle-point systems with vector of several unknowns. In this paper this vector is composed of pressure and its gradient and the saddle-point system is the mixed formulation of the Darcy problem.

We study approximations of eigenvalue problems for integral operators associated with kernel functions of exponential type. We show convergence rate | λ k - λ k , h | ≤ C k ⁢ h 2 {\lvert\lambda_{k}-\lambda_{k,h}\rvert\leq C_{k}h^{2}} in the case of lowest order approximation for both Galerkin and Nyström methods, where h is the mesh size, λ k {\lambda_{k}} and λ k , h {\lambda_{k,h}} are the exact and approximate kth largest eigenvalues, respectively. We prove that the two methods are numerically equivalent in the sense that | λ k , h ( G ) - λ k , h ( N ) | ≤ C ⁢ h 2 {|\lambda^{(G)}_{k,h}-\lambda^{(N)}_{k,h}|\leq Ch^{2}} , where λ k , h ( G ) {\lambda^{(G)}_{k,h}} and λ k , h ( N ) {\lambda^{(N)}_{k,h}} denote the kth largest eigenvalues computed by Galerkin and Nyström methods, respectively, and C is a eigenvalue independent constant. The theoretical results are accompanied by a series of numerical experiments.

This chapter deals with the scope and limitations of lumped parameter models. Numerical methods for heart model are also discussed. The model is applied to simulate aortic insufficiency and mitral valve stenosis.

In this chapter, the hydrogeological multi-physics models and corresponding numerical methods are presented basing on the authors’ experience of GeRa hydrogeological code [87, 93] development and its applications. Flow in unsaturated conditions, reactive transport, and density-driven flow models are addressed.

This chapter introduces 1D hemodynamic equations in a single vessel and their extension to a network of vessels through boundary and coupling conditions of different types. Advanced approaches use 1D hemodynamic models as a part of multiscale modeling. 0D–1D, 3D–1D, and 3D–0D–1D multiscale models are described. Numerical methods are also discussed. Modifications of reduced order models, which account for physiological conditions and some pathologies, are presented.

Transcranial ultrasound examination is hampered by the skull which acts as an irregular aberrator of the ultrasound signal. Numerical recovery of the ultrasound field can help in elimination of aberrations induced by the skull. In this paper, we address the simulation of medical phantom scanning through silicon aberrators with wave notching. The numerical model is based on the 2D acoustic equations which are solved by the wavefront construction raytracing method. Numerical B-scan images are compared with experimental B-scan images.

This paper studies finite element discretizations for three types of time-dependent PDEs, namely heat equation, scalar conservation law and wave equation, which we reformulate as first order systems in a least-squares setting, subject to a space-time conservation constraint (coming from the original PDE). Available piecewise polynomial finite element spaces in ( n + 1 ) -dimensions for functional spaces from the ( n + 1 ) -dimensional de Rham sequence for n = 2 , 3 are used for the implementation of the method. Computational results illustrating the error behavior, iteration counts and performance of block-diagonal and monolithic geometric multigrid preconditioners are presented for the discrete CFOSLS system. The results are obtained from a parallel implementation of the methods for which we report reasonable scalability.

The paper proposes a stable time-splitting method for the numerical simulation of free-surface viscous flows. The key features of the method are a semi-Lagrangian scheme for the level-set function transport improved with MacCormack predictor–corrector step with limiting strategy and an adaptive volume-correction procedure. The spatial discretization is done by a hybrid finite volume/finite difference method on dynamically adaptive hexahedral meshes. Numerical verification is done by comparing full-scale 3D numerical simulations of the sloshing tank and the coastal wave run-up with other numerical and experimental results known from the literature.

The paper develops a hybrid method for solving a system of advection–diffusion equations in a bulk domain coupled to advection–diffusion equations on an embedded surface. A monotone nonlinear finite volume method for equations posed in the bulk is combined with a trace finite element method for equations posed on the surface. In our approach, the surface is not fitted by the mesh and is allowed to cut through the background mesh in an arbitrary way. Moreover, a triangulation of the surface into regular shaped elements is not required. The background mesh is an octree grid with cubic cells. As an example of an application, we consider the modeling of contaminant transport in fractured porous media. One standard model leads to a coupled system of advection–diffusion equations in a bulk (matrix) and along a surface (fracture). A series of numerical experiments with both steady and unsteady problems and different embedded geometries illustrate the numerical properties of the hybrid approach. The method demonstrates great flexibility in handling curvilinear or branching lower dimensional embedded structures.

The paper introduces a finite element method for the Navier-Stokes equations of incompressible viscous fluid in a time-dependent domain. The method is based on a quasi-Lagrangian formulation of the problem and handling the geometry in a time-explicit way. We prove that numerical solution satisfies a discrete analogue of the fundamental energy estimate. This stability estimate does not require a CFL time-step restriction. The method is further applied to simulation of a flow in a model of the left ventricle of a human heart, where the ventricle wall dynamics is reconstructed from a sequence of contrast enhanced Computed Tomography images.

The paper develops a method for the numerical simulation of a free-surface flow of incompressible viscous fluid around a streamlined body. The body is a rigid stationary construction partially submerged in the fluid. The application we are interested in the paper is a flow around a surface mounted offshore oil platform. The numerical method builds on a hybrid finite volume / finite difference discretization using adaptive octree cubic meshes. The mesh is dynamically refined towards the free surface and the construction. Special care is taken to devise a discretization for the case of curvilinear boundaries and interfaces immersed in the octree Cartesian background computational mesh. To demonstrate the accuracy of the method, we show the results for two benchmark problems: the sloshing 3D container and the channel laminar flow passing the 3D cylinder of circular cross-section. Further, we simulate numerically a flow with surface waves around an offshore oil platform for the realistic set of geophysical data.

The systems of monodomain and bidomain equations are widely used in modeling of cardiac electrophysiology. The bidomain problem may be extended with an addition of electrically conductive bath to simulate the conductance of the human body. Recently, there has been a growing interest in developing numerical methods for these systems. They can be used in various clinical applications. The numerical solution of monodomain and bidomain equations on high-resolution meshes is computationally expensive in 3D. Thereby the corresponding computational frameworks should take advantage of High Performance Computing (HPC) architectures. Another challenge is to verify that a given code provides the correct numerical solution of the governing equations. In this work we present our approach for the solution of monodomain and bidomain equations based on Ani3D (Advanced Numerical Instruments) framework (Advanced Numerical Instruments 3D. https://sourceforge.net/p/ani3d). We use finite element method on unstructured tetrahedral meshes for solving PDEs of the diffusion type. To calculate the ionic current density from a system of ODEs, we use CVODE software (Hindmarsh et al., ACM Trans Math Softw 31:363–396, 2005) and CellML repository (Yu et al., Bioinformatics 27:743–744, 2011). Verification of the computational model is conducted by several benchmarks. In order to parallelize the computation of the ionic currents in each node of the computational grid, we use the OpenMP technology. One of the advantages of the Ani3D framework is that it provides tools for usage of anisotropic grids and tensors, which are crucial in real heart modeling due to the essential anisotropic features of the myocardial tissue.

The paper studies an iterative method for algebraic problems arising in numerical simulation of blood flows. Here we focus on a numerical solver for the fluid part of otherwise coupled fluid-structure system of equations which models the hemodynamics in vessels. Application of the finite element method and semi-implicit time discretization leads to the discrete Oseen problem at every time step of the simulation. The problem challenges numerical methods by anisotropic geometry, open boundary conditions, small time steps and transient flow regimes. We review known theoretical results and study the performance of recently proposed preconditioners based on two-parameter threshold ILU factorization of non-symmetric saddle point problems. The preconditioner is applied to the linearized Navier–Stokes equations discretized by the stabilized Petrov–Galerkin finite element (FE) method. Careful consideration is given to the dependence of the solver on the stabilization parameters of the FE method. We model the blood flow in the digitally reconstructed right coronary artery under realistic physiological regimes. The paper discusses what is special in such flows for the iterative algebraic solvers, and shows how the two-parameter ILU preconditioner is able to meet these specifics.

The paper is devoted to verification of previously proposed technique of medical ultrasound modelling [

We generalize the construction and analysis of auxiliary space preconditioners to the n-dimensional finite element subcomplex of the de Rham complex. These preconditioners are based on a generalization of a decomposition of Sobolev space functions into a regular part and a potential. A discrete version is easily established using the tools of finite element exterior calculus. We then discuss the four-dimensional de Rham complex in detail. By identifying forms in four dimensions (4D) with simple proxies, form operations are written out in terms of familiar algebraic operations on matrices, vectors, and scalars. This provides the basis for our implementation of the preconditioners in 4D. Extensive numerical experiments illustrate their performance, practical scalability, and parameter robustness, all in accordance with the theory.

The paper studies a splitting method for the numerical time-integration of the system of partial differential equations describing the motion of viscous incompressible fluid with free boundary subject to surface tension forces. The method splits one time step into a semi-Lagrangian treatment of the surface advection and fluid inertia, an implicit update of viscous terms and the projection of velocity into the subspace of divergence-free functions. We derive several conservation properties of the method and a suitable energy estimate for numerical solutions. Under certain assumptions on the smoothness of the free surface and its evolution, this leads to a stability result for the numerical method. Efficient computations of free surface flows of incompressible viscous fluids need several other ingredients, such as dynamically adapted meshes, surface reconstruction and level set function re-initialization. These enabling techniques are discussed in the paper as well. The properties of the method are illustrated with a few numerical examples. These examples include analytical tests and the oscillating droplet benchmark problem.

The Brinkman model is a unified law governing the flow of a viscous fluid in an inhomogeneous medium, where fractures, bubbles, or channels alternate inside a porous matrix. In this work, we explore a novel mixed formulation of the Brinkman problem based on the Hodge decomposition of the vector Laplacian. Introducing the flow's vorticity as an additional unknown, this formulation allows for a uniformly stable and conforming discretization by standard finite elements (Nedelec, Raviart--Thomas, piecewise discontinuous). A priori error estimates for the discretization error in the $H({\rm curl}; \Omega)-H({\rm div}; \Omega)-L^2(\Omega)$ norm of the solution, which are optimal with respect to the approximation properties of finite element spaces, are obtained. The theoretical results are illustrated with numerical experiments. Finally, the proposed formulation allows for a scalable block diagonal preconditioner which takes advantage of the auxiliary space algebraic multigrid solvers for $H({\rm curl})$ and $H...

We present a new nonlinear monotone finite volume method for diffusion equation and its application to two-phase flow model. We consider full anisotropic discontinuous diffusion or permeability tensors on conformal polyhedral meshes. The approximation of the diffusive flux uses the nonlinear two-point stencil which provides the conventional seven-point stencil for the discrete diffusion operator on cubic meshes. We show that the quality of the discrete flux in a reservoir simulator has great effect on the front behavior and the water breakthrough time. We compare two two-point flux approximations (TPFA), the proposed nonlinear TPFA and the conventional linear TPFA, and multipoint flux approximation (MPFA). The new nonlinear scheme has a number of important advantages over the traditional linear discretizations. Compared to the linear TPFA, the nonlinear TPFA demonstrates low sensitivity to grid distortions and provides appropriate approximation in case of full anisotropic permeability tensor. For nonorthogonal grids or full anisotropic permeability tensors, the conventional linear TPFA provides no approximation, while the nonlinear flux is still first-order accurate. The computational work for the new method is higher than the one for the conventional TPFA, yet it is rather competitive. Compared to MPFA, the new scheme provides sparser algebraic systems and thus is less computational expensive. Moreover, it is monotone which means that the discrete solution preserves the nonnegativity of the differential solution.

The INMOST mathematical modeling toolkit helps a user to formulate and solve a problem of partial differential equations on general meshes in parallel. The current work covers: data structure description for efficient distributed unstructured mesh representation, interrelation of mesh elements with maximal flexibility of supported types of the mesh, treatment of ghost cells and distribution of mesh data for parallel execution, flexible templates for the implementation of numerical schemes, convenient framework for parallel linear systems assembly and solution. We also present aspects of the implementation and a simple example of application of INMOST to the solution of anisotropic diffusion problem. On this example we demonstrate the application of INMOST for all the stages of numerical modeling: construction of the distributed mesh, assignment of the problem data to the elements, problem discretization on local domain, solution of linear system in parallel. INMOST is a newly developed, flexible and efficient numerical analysis framework that provides scientists the infrastructure for designing highly scalable high performance applications for mathematical modeling.

Correct diagnostics of vascular pathologies underlies treatment success for patients with cerebrovascular diseases. Transcranial ultrasound is the well-known method for diagnostic of cerebrovascular diseases. Despite high sensitivity and specificity of the method, transcranial ultrasound has some limitations related to the B-mode image quality and accurate insonation of vessels of interest. Overcoming these limitations enables to enhance the quality of the diagnostic procedure. The present work addresses the numerical simulation of ultrasound propagation in a human head by a grid-characteristic method. We used a human tissue-mimicking phantom to verify our numerical model in terms of the accuracy of distance estimation. We obtained pressure distributions within a 3D segmented model of a human head. Our pilot study has some limitations, nevertheless the simulation results demonstrate that mathematical modelling of the transcranial ultrasound can be an effective tool to enhance the ultrasound examination.

We study regular decompositions for H(div) spaces. In particular, we show that such regular decompositions are closely related to a previously studied ``inf-sup'' condition for parameter-dependent Stokes problems, for which we provide an alternative, more direct, proof.

In this paper we generalize the analysis of classical multigrid and two-level overlapping Schwarz methods for 2nd order elliptic boundary value problems to problems with large discontinuities in the coefficients that are not resolved by the coarse grids or the subdomain partition. The theoretical results provide a recipe for designing hierarchies of standard piecewise linear coarse spaces such that the multigrid convergence rate and the condition number of the Schwarz preconditioned system do not depend on the coefficient variation or on any mesh parameters. An assumption we have to make is that the coarse grids are sufficiently fine in the vicinity of cross points or where regions with large diffusion coefficients are separated by a narrow region where the coefficient is small. We do not need to align them with possible discontinuities in the coefficients. The proofs make use of novel stable splittings based on weighted quasi-interpolants and weighted Poincaré-type inequalities. Numerical experiments are included that illustrate the sharpness of the theoretical bounds and the necessity of the technical assumptions.

We derive defect correction scheme for constructing the sequence of polynomials of best approximation in the uniform norm to 1/x on a finite interval with positive endpoints. As an application, we consider two-level methods for scalar elliptic partial differential equation (PDE), where the relaxation on the fine grid uses the aforementioned polynomial of best approximation. Based on a new smoothing property of this polynomial smoother that we prove, combined with a proper choice of the coarse space, we obtain as a corollary, that the convergence rate of the resulting two-level method is uniform with respect to the mesh parameters, coarsening ratio and PDE coefficient variation.

In this paper we study a numerical method for the simulation of free surface ows of viscoplastic (Herschel-Bulkley) uids. The approach is based on the level set method for capturing the free surface evolution and on locally rened and dynamically adapted octree cartesian staggered grids for the discretization of uid and level set equations. A regularized model is applied to handle the non-dierentiabil ity of the constitutive relations. We consider an extension of the stable approximation of the Newtonian ow equations on staggered grid to approximate the viscoplastic model and level-set equations if the free boundary evolves and the mesh is dynamically rened or coarsened. The numerical method is rst validated for a Newtonian case. In this case, the convergence of numerical solutions is observed towards experimental data when the mesh is rened. Further we compute several 3D viscoplastic Herschel-Bulkley uid ows over incline planes for the

We describe a new method for generating meshes that minimize the gradient of a discretization error. The key element of this method is construction of a tensor metric from edge-based error estimates. In our papers [1-4] we applied this metric for generating meshes that minimize the gradient of P"1-interpolation error and proved that for a mesh with N triangles, the L^2-norm of gradient of the interpolation error is proportional to N^-^1^/^2. In the present paper we recover the tensor metric using hierarchical a posteriori error estimates. Optimal reduction of the discretization error on a sequence of adaptive meshes will be illustrated numerically for boundary value problems ranging from a linear isotropic diffusion equation to a nonlinear transonic potential equation.

The distributed relaxation method for the Stokes problem has been advertised as an adequate change of variables that leads to a lower triangular system with Laplace operators on the main diagonal for which multigrid methods are very efficient. We show that under high regularity of the Laplacian, the transformed system admits almost block-lower triangular form. We analyze the distributed relaxation method and compare it with other iterative methods for solving the Stokes system. We also present numerical experiments illustrating the effectiveness of the transformation which is well established for certain finite difference discretizations of Stokes problems. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 000: 000–000, 2010

International audience; A heterogeneous domain decomposition with non-matching grids is developed, extending the Aitken-Schwarz method [M. Garbey, D. Tromeur-Dervout, On some Aitken like acceleration of the Schwarz method, Internat. J. Numer. Methods Fluids 40 (2002) 1493- 1513] which has proved to be efficient on metacomputing architectures. This novel numerical technique for the approximate solution of boundary value problems applies when the solution is assumed to possess many features in several subdomains such as in underground environmental problems with different geological layers. The numerical technique involves methods for the approximation and the solution of arising linear systems, as well as for parallel computing issues. We consider a natural coupling between the mixed finite element and spectral element approximations as well as the efficient solution of the coupled discrete systems on remote parallel computers with different architectures connected via a low speed network.

We present a new second-order accurate monotone finite volume (FV) method for the steady-state advection-diffusion equation. The method uses a nonlinear approximation for both diffusive and advective fluxes and guarantees solution non-negativity. The interpolation-free approximation of the diffusive flux uses the nonlinear two-point stencil proposed in Lipnikov [23]. Approximation of the advective flux is based on the second-order upwind method with a specially designed minimal nonlinear correction. The second-order convergence rate and monotonicity are verified with numerical experiments.

This report proposes a new class of interpolation procedures for use in algebraic multigrid (AMG). The procedure is general in that it applies to s.p.d. finite element discretization problems that include scalar and vector elliptic PDEs, and it can be adapted to the time-domain Maxwell (definite $H(\mathrm{curl})$) problems. It can also be viewed as an extension of previously proposed vector-fitting interpolation procedures and can be used in adaptive AMG cycles. We illustrate the performance of the new interpolation matrices on a number of test problems.

In this article, we develop and analyze a mixed finite element method for the Stokes equations. Our mixed method is based on the pseudostress-velocity formulation. The pseudostress is approximated by the Raviart-Thomas (RT) element of index k ≥ 0 and the velocity by piecewise discontinuous polynomials of degree k. It is shown that this pair of finite elements is stable and yields quasi-optimal accuracy. The indefinite system of linear equations resulting from the discretization is decoupled by the penalty method. The penalized pseudostress system is solved by the H(div) type of multigrid method and the velocity is then calculated explicitly. Alternative preconditioning approaches that do not involve penalizing the system are also discussed. Finally, numerical experiments are presented. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

We developed a new monotone finite volume method for diffusion equations. The second-order linear methods, such as the multipoint flux approximation, mixed finite element and mimetic finite difference methods, are not monotone on strongly anisotropic meshes or for diffusion problems with strongly anisotropic coefficients. The finite volume (FV) method with linear two-point flux approximation is monotone but not even first-order accurate in these cases. The developed monotone method is based on a nonlinear two-point flux approximation. It does not require any interpolation scheme and thus differs from other nonlinear finite volume methods based on a two-point flux approximation. The second-order convergence rate is verified with numerical experiments.

We describe a domain decomposition algorithm for use in several variants of the parallel adaptive meshing paradigm of Bank and Holst. This algorithm has low communication, makes extensive use of existing sequential solvers, and exploits in several important ways data generated as part of the adaptive meshing paradigm. We show that for an idealized version of the algorithm, the rate of convergence is independent of both the global problem size N and the number of subdomains p used in the domain decomposition partition. Numerical examples illustrate the effectiveness of the procedure.

We consider a non-linear finite volume (FV) scheme for stationary diffusion equation. We prove that the scheme is monotone, i.e. it preserves positivity of analytical solutions on arbitrary triangular meshes for strongly anisotropic and heterogeneous full tensor coefficients. The scheme is extended to regular star-shaped polygonal meshes and isotropic heterogeneous coefficients.

We consider solution techniques for the coupling of Darcy and Stokes flow problems. The study was motivated by the simulation of the interaction between channel flow and subsurface water flow for realistic data and arbitrary interfaces between the two different flow regimes. Here, the emphasis is on the efficient iterative solution of the coupled problem based on efficient solvers for the discrete Stokes and Darcy problems.

Patient-specific simulations of human physiological processes remain the challenge for many years. Detailed 3D reconstruction of body anatomical parts on the basis of medical images is an important stage of individualized simulations in physiology. In this paper we present and develop the methods and algorithms for construction of patient-specific discrete geometric models. These models are represented by anatomically correct computational meshes. Practical use of these methods is demonstrated for two important medical applications: numerical evaluation of fractional flow reserve in coronary arteries and electrocardiography simulation

The paper develops a semi-Lagrangian method for the numerical integration of the transport equation discretized on adaptive Cartesian cubic meshes. We use dynamically adaptive graded Cartesian grids. They allow for a fast grid reconstruction in the course of numerical integration. The suggested semi- Lagrangian method uses a higher order interpolation with a limiting strategy and a back-and-forth correction of the numerical solution. The interpolation operators have compact nodal stencils. In a series of experiments with dynamically adapted meshes, we demonstrate that the method has at least the second-order convergence and acceptable conservation and monotonicity properties.

This work is the numerical study of the age-dependent responses of the vascular system under low-mass high-speed impact scenario. The grid-characteristic method on the adaptive mesh model of the human thorax is the numerical tool of the study. Due to the lack of valid vascular injury criteria, the numerical model only provides information on injury risk. The numerical simulation demonstrates that an older age changes significantly the vascular response and increases the risk of aorta injury. We focused on the aorta because its rupture is the general consequence of vehicle accidents (great mass impacts at relatively low velocity). Our numerical results are in good agreement with previous studies of great-mass low-speed blunt thorax impact.

We present a theory for algebraic multigrid (AMG) methods that allows for general smoothing processes and general coarsening approaches. The goal of the theory is to provide guidance in the development of new, more robust, AMG algorithms. In particular, we introduce several compatible relaxation methods and give theoretical justification for their use as tools for measuring the quality of coarse grids.

We consider an interior penalty discontinuous approximation for symmetric elliptic problems of second order on non-matching grids in this paper. The main result is an almost optimal error estimate for the interior penalty approximation of the original problem based on partitioning of the domain into a finite number of subdomains. Further, an error analysis for the finite element approximation of the penalty formulation is given. Finally, numerical experiments on a series of model second order problems are presented.

We discuss several fully implicit techniques for solving the nonlinear algebraic system arising in an expanded mixed finite element or cell-centered finite difference discretization of two- and three-phase porous media flow. Every outer nonlinear Newton iteration requires solution of a nonsymmetric Jacobian linear system. Two major types of preconditioners, supercoarsening multigrid (SCMG) and two-stage, are developed for the GMRES iteration applied to the solution of the Jacobian system. The SCMG reduces the three-dimensional system to two dimensions using a vertical aggregation followed by a two-dimensional multigrid. The two-stage preconditioners are based on decoupling the system into a pressure and concentration equations. Several pressure preconditioners of different types are described. Extensive numerical results are presented using the integrated parallel reservoir simulator (IPARS) and indicate that these methods have low arithmetical complexity per iteration and good convergence rates.