characteristic cut finite element methods for convection diffusion problems on time dependent surfaces we develop a finite element method for convection diffusion problems on a given time dependent surface for instance modeling the evolution of a surfactant the method is based on a characteristic

a hybrid finite volume finite element method for bulk surface coupled problems abstract 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

full gradient stabilized cut finite element methods for surface partial differential equations we propose and analyze a new stabilized cut finite element method for the laplace beltrami operator on a closed surface the new stabilization term provides control of the full r 3 gradient on the

cut finite element methods for coupled bulk surface problems we develop a cut finite element method for a second order elliptic coupled bulk surface model problem we prove a priori estimates for the energy and l 2 l2 norms of the error using stabilization terms we show that the resulting algebraic system of equations has a similar condition number as a standard fitted finite element method finally we present a numerical example illustrating the accuracy and the robustness of our approach

tree methods for moving interfaces fast adaptive numerical methods for solving moving interface problems are presented the methods combine a level set approach with frequent redistancing and semi lagrangian time stepping schemes which are explicit yet unconditionally stable a quadtree mesh is used to concentrate computational effort on the interface so the methods move an interface withndegrees of freedom ino nlogn work per time step efficiency is increased by taking large time steps even for parabolic curvature flows the methods compute accurate viscosity solutions to a wide variety of difficult moving interface problems involving merging anisotropy faceting and curvature

an adaptive finite element method for the laplace beltrami operator on implicitly defined surfaces we present an adaptive finite element method for approximating solutions to the laplace beltrami equation on surfaces in mathbb r 3 which may be implicitly represented as level sets of smooth functions residual type a posteriori error bounds which show that the error may be split into a residual part and a geometric part are established in addition implementation issues are discussed and several computational examples are given

modeling and computation of two phase geometric biomembranes using surface finite elements biomembranes consisting of multiple lipids may involve phase separation phenomena leading to coexisting domains of different lipid compositions the modeling of such biomembranes involves an elastic or bending energy together with a line energy associated with the phase interfaces this leads to a free boundary problem for the phase interface on the unknown equilibrium surface which minimizes an energy functional subject to volume and area constraints in this paper we propose a new computational tool for computing equilibria based on an l 2 relaxation flow for the total energy in which the line energy is approximated by a surface ginzburg landau phase field functional the relaxation dynamics couple a nonlinear fourth order geometric evolution equation of willmore flow type for the membrane with a surface allen cahn equation describing the lateral decomposition a novel system is derived involving second order elliptic operators where the field variables are the positions of material points of the surface the mean curvature vector and the surface phase field function the resulting variational formulation uses h 1 spaces and we employ triangulated surfaces and h 1 conforming quadratic surface finite elements for approximating solutions together with a semi implicit time discretization of the evolution equations an iterative scheme is obtained essentially requiring linear solvers only numerical experiments are presented which exhibit convergence and the power of this new method for two component geometric biomembranes by computing equilibria such as dumbbells discocytes and starfishes with lateral phase separation

a posteriori error estimators for convection diffusion equations we derive a posteriori error estimators for convection diffusion equations with dominant convection the estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds only depends on the local mesh peclet number the estimators are either based on the evaluation of local residuals or on the solution of discrete local dirichlet or neumann problems

error analysis of a space time finite element method for solving pdes on evolving surfaces in this paper we present an error analysis of an eulerian finite element method for solving parabolic partial differential equations pdes posed on evolving hypersurfaces in rd d 2 3 the method employs discontinuous piecewise linear in time continuous piecewise linear in space finite elements and is based on a space time weak formulation of a surface pde problem trial and test surface finite element spaces consist of traces of standard volumetric elements on a space time manifold resulting from the evolution of a surface we prove first order convergence in space and time of the method in an energy norm and second order convergence in a weaker norm furthermore we derive regularity results for solutions of parabolic pdes on an evolving surface which we need in a duality argument used in the proof of the second order convergence estimate

an adaptive octree finite element method for pdes posed on surfaces abstract the paper develops a finite element method for partial differential equations posed on hypersurfaces in r n n 2 3 the method uses traces of bulk finite element functions on a surface embedded in a volumetric domain the bulk finite element space is defined on an octree grid which is locally refined or coarsened depending on error indicators and estimated values of the surface curvatures the cartesian structure of the bulk mesh leads to easy and efficient adaptation process while the trace finite element method makes fitting the mesh to the surface unnecessary the number of degrees of freedom involved in computations is consistent with the two dimension nature of surface pdes no parametrization of the surface is required it can be given implicitly by a level set function in practice a variant of the marching cubes method is used to recover the surface with the second order accuracy we prove the optimal order of accuracy for the trace finite element method in h 1 and l 2 surface norms for a problem with smooth solution and quasi uniform mesh refinement experiments with less regular problems demonstrate optimal convergence with respect to the number of degrees of freedom if grid adaptation is based on an appropriate error indicator the paper shows results of numerical experiments for a variety of geometries and problems including advection diffusion equations on surfaces analysis and numerical results of the paper suggest that combination of cartesian adaptive meshes and the unfitted trace finite elements provide simple efficient and reliable tool for numerical treatment of pdes posed on surfaces

an eulerian space time finite element method for diffusion problems on evolving surfaces in this paper we study numerical methods for the solution of partial differential equations on evolving surfaces the evolving hypersurface in bbb r d defines a d dimensional space time manifold in the space time continuum bbb r d 1 we derive and analyze a variational formulation for a class of diffusion problems on the space time manifold for this variational formulation new well posedness and stability results are derived the analysis is based on an inf sup condition and involves some natural but nonstandard anisotropic function spaces based on this formulation a discrete in time variational formulation is introduced that is very suitable as a starting point for a discontinuous galerkin dg space time finite element discretization this dg space time method is explained and results of numerical experiments are presented that illustrate its properties

eulerian finite element methods for parabolic equations on moving surfaces three new eulerian finite element methods for parabolic pdes on a moving surface gamma t are presented and compared in numerical experiments these are space time galerkin methods which are derived from a weak formulation in space and time the trial and test spaces contain the traces on the space time manifold of an outer prismatic finite element space the numerical experiments show that two of the methods converge with second order with respect to both the time step size and the spatial mesh width

variational problems and partial differential equations on implicit surfaces a novel framework for solving variational problems and partial differential equations for scalar and vector valued data defined on surfaces is introduced in this paper the key idea is to implicitly represent the surface as the level set of a higher dimensional function and to solve the surface equations in a fixed cartesian coordinate system using this new embedding function the equations are then both intrinsic to the surface and defined in the embedding space this approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces as is commonly done in the literature we describe the framework and present examples in computer graphics and image processing applications including texture synthesis flow field visualization and image and vector field intrinsic regularization for data defined on 3d surfaces

a stabilized cut finite element method for partial differential equations on surfaces the laplace beltrami operator we consider solving the laplace beltrami problem on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra the mesh does not respect the surface and thus t

gerris a tree based adaptive solver for the incompressible euler equations in complex geometries an adaptive mesh projection method for the time dependent incompressible euler equations is presented the domain is spatially discretised using quad octrees and a multilevel poisson solver is used to obtain the pressure complex solid boundaries are represented using a volume of fluid approach second order convergence in space and time is demonstrated on regular statically and dynamically refined grids the quad octree discretisation proves to be very flexible and allows accurate and efficient tracking of flow features the source code of the method implementation is freely available

unfitted finite element methods using bulk meshes for surface partial differential equations in this paper we define new unfitted finite element methods for numerically approximating the solution of surface partial differential equations using bulk finite elements the key idea is that the n dimensional hypersurface gamma subset mathbb r n 1 is embedded in a polyhedral domain in mathbb r n 1 consisting of a union mathscr t h of n 1 simplices the unifying feature of the methodological approach is that the finite element approximating space is based on continuous piecewise linear finite element functions on the bulk triangulation mathscr t h which is independent of gamma our first method is a sharp interface method sif which uses the bulk finite element space in an approximating weak formulation obtained from integration on a polygonal approximation gamma h of gamma the full gradient is used rather than the projected tangential gradient and it is this which distinguishes sif from the method of m a olshanskii a reusken and j grande siam j nu

a test for partial differential expression even in a single tissue type cancer is often a collection of different diseases each with its own genetic mechanism consequently a gene may be expressed in some but not all of the tissues in a s

an adaptive surface finite element method based on volume meshes in this paper we define an adaptive version of a recently introduced finite element method for numerical treatment of elliptic pdes defined on surfaces the method makes use of a standard outer volume mesh to discretize an equation on a two dimensional surface embedded in mathbb r 3 extension of the equation from the surface is avoided but the number of degrees of freedom d o f is optimal in the sense that it is comparable to methods in which the surface is meshed directly in previous work it was proved that the method exhibits optimal order of convergence for an elliptic surface pde if the volume mesh is uniformly refined in this paper we extend the method and develop an a posteriori error analysis which admits adaptively refined meshes the reliability of a residual type a posteriori error estimator is proved and both reliability and efficiency of the estimator are studied numerically in a series of experiments a simple adaptive refinement strategy based on the error estimator is numerical

a finite element method for surface pdes matrix properties we consider a recently introduced new finite element approach for the discretization of elliptic partial differential equations on surfaces the main idea of this method is to use finite element spaces that are induced by triangulations of an outer domain to discretize the partial differential equation on the surface the method is particularly suitable for problems in which there is a coupling with a problem in an outer domain that contains the surface for example two phase flow problems it has been proved that the method has optimal order of convergence both in the h 1 and in the l 2 norm in this paper we address linear algebra aspects of this new finite element method in particular the conditioning of the mass and stiffness matrix is investigated for the two dimensional case we present an analysis which proves that the effective spectral condition number of the diagonally scaled mass matrix and the diagonally scaled stiffness matrix behaves like h 3 ln h and h 2 ln h respectively where h is the mesh size of the outer triangulation

asymptotics frequency modulation and low regularity ill posedness for canonical defocusing equations in a recent paper kenig ponce and vega study the low regularity behavior of the focusing nonlinear schrodinger nls focusing modified korteweg de vries mkdv and complex korteweg de vries kdv equations using soliton and breather solutions they demonstrate the lack of local well posedness for these equations below their respective endpoint regularities in this paper we study the defocusing analogues of these equations namely defocusing nls defocusing mkdv and real kdv all in one spatial dimension for which suitable soliton and breather solutions are unavailable we construct for each of these equations classes of modified scattering solutions which exist globally in time and are asymptotic to solutions of the corresponding linear equations up to explicit phase shifts these solutions are used to demonstrate lack of local well posedness in certain sobolev spaces in the sense that the dependence of solutions upon initial data fails to be uniformly continuous in particular we show that the mkdv flow is not uniformly continuous in the l 2 topology despite the existence of global weak solutions at this regularity finally we investigate the kdv equation at the endpoint regularity h and construct solutions for both the real and complex kdv equations the construction provides a nontrivial time interval t t and a locally lipschitz continuous map taking the initial data in h to a distributional solution u c 0 t t h which is uniquely defined for all smooth data the proof uses a generalized miura transform to transfer the existing endpoint regularity theory for mkdv to kdv

a finite element method for elliptic equations on surfaces in this paper a new finite element approach for the discretization of elliptic partial differential equations on surfaces is treated the main idea is to use finite element spaces that are induced by triangulations of an outer domain to discretize the partial differential equation on the surface the method is particularly suitable for problems in which there is a coupling with a flow problem in an outer domain that contains the surface we give an analysis that shows that the method has optimal order of convergence both in the h 1 and in the l 2 norm results of numerical experiments are included that confirm this optimality

an accurate adaptive solver for surface tension driven interfacial flows a method combining an adaptive quad octree spatial discretisation geometrical volume of fluid interface representation balanced force continuum surface force surface tension formulation and height function curvature estimation is presented the extension of these methods to the quad octree discretisation allows adaptive variable resolution along the interface and is described in detail the method is shown to recover exact equilibrium to machine accuracy between surface tension and pressure gradient in the case of a stationary droplet irrespective of viscosity and spatial resolution accurate solutions are obtained for the classical test case of capillary wave oscillations an application to the capillary breakup of a jet of water in air further illustrates the accuracy and efficiency of the method the source code of the implementation is freely available as part of the gerris flow solver