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Erik Burman

Publications and source records attributed to Erik Burman.

At least 91 records · Page 5Linked to original sources
arXiv2019

Dirichlet Boundary Value Correction using Lagrange Multipliers

We propose a boundary value correction approach for cases when curved boundaries are approximated by straight lines (planes) and Lagrange multipliers are used to enforce Dirichlet boundary conditions. The approach allows for optimal order convergence for polynomial order up to 3. We show the relation to the Taylor series expansion approach used by Bramble, Dupont and Tomée [Math. Comp., 26:869--879, 1972] in the context of Nitsche's method and, in the case of inf-sup stable multiplier methods, prove a priori error estimates with explicit dependence on the meshsize and distance between the exact and approximate boundary.

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arXiv2019

Cut Topology Optimization for Linear Elasticity with Coupling to Parametric Nondesign Domain Regions

We develop a density based topology optimization method for linear elasticity based on the cut finite element method. More precisely, the design domain is discretized using cut finite elements which allow complicated geometry to be represented on a structured fixed background mesh. The geometry of the design domain is allowed to cut through the background mesh in an arbitrary way and certain stabilization terms are added in the vicinity of the cut boundary, which guarantee stability of the method. Furthermore, in addition to standard Dirichlet and Neumann conditions we consider interface conditions enabling coupling of the design domain to parts of the structure for which the design is already given. These given parts of the structure, called the nondesign domain regions, typically represents parts of the geometry provided by the designer. The nondesign domain regions may be discretized independently from the design domains using for example parametric meshed finite elements or isogeometric analysis. The interface and Dirichlet conditions are based on Nitsche's method and are stable for the full range of density parameters. In particular we obtain a traction-free Neumann condition in the limit when the density tends to zero.

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arXiv2018

Boundary element methods with weakly imposed boundary conditions

We consider boundary element methods where the Calderón projector is used for the system matrix and boundary conditions are weakly imposed using a particular variational boundary operator designed using techniques from augmented Lagrangian methods. Regardless of the boundary conditions, both the primal trace variable and the flux are approximated. We focus on the imposition of Dirichlet, mixed Dirichlet--Neumann, and Robin conditions. A salient feature of the Robin condition is that the conditioning of the system is robust also for stiff boundary conditions. The theory is illustrated by a series of numerical examples.

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arXiv2018

A finite element data assimilation method for the wave equation

We design a primal-dual stabilized finite element method for the numerical approximation of a data assimilation problem subject to the acoustic wave equation. For the forward problem, piecewise affine, continuous, finite element functions are used for the approximation in space and backward differentiation is used in time. Stabilizing terms are added on the discrete level. The design of these terms is driven by numerical stability and the stability of the continuous problem, with the objective of minimizing the computational error. Error estimates are then derived that are optimal with respect to the approximation properties of the numerical scheme and the stability properties of the continuous problem. The effects of discretizing the (smooth) domain boundary and other perturbations in data are included in the analysis.

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arXiv2018

Primal dual mixed finite element methods for indefinite advection--diffusion equations

We consider primal-dual mixed finite element methods for the advection--diffusion equation. For the primal variable we use standard continuous finite element space and for the flux we use the Raviart-Thomas space. We prove optimal a priori error estimates in the energy- and the $L^2$-norms for the primal variable in the low Peclet regime. In the high Peclet regime we also prove optimal error estimates for the primal variable in the $H(div)$ norm for smooth solutions. Numerically we observe that the method eliminates the spurious oscillations close to interior layers that pollute the solution of the standard Galerkin method when the local Peclet number is high. This method, however, does produce spurious solutions when outflow boundary layer presents. In the last section we propose two simple strategies to remove such numerical artefacts caused by the outflow boundary layer and validate them numerically.

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arXiv2018

Hybridized CutFEM for Elliptic Interface Problems

We design and analyze a hybridized cut finite element method for elliptic interface problems. In this method very general meshes can be coupled over internal unfitted interfaces, through a skeletal variable, using a Nitsche type approach. We discuss how optimal error estimates for the method are obtained using the tools of cut finite element methods and prove a condition number estimate for the Schur complement. Finally, we present illustrating numerical examples.

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arXiv2018

Unique continuation for the Helmholtz equation using stabilized finite element methods

In this work we consider the computational approximation of a unique continuation problem for the Helmholtz equation using a stabilized finite element method. First conditional stability estimates are derived for which, under a convexity assumption on the geometry, the constants grow at most linearly in the wave number. Then these estimates are used to obtain error bounds for the finite element method that are explicit with respect to the wave number. Some numerical illustrations are given.

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arXiv2018

A Stable Cut Finite Element Method for Partial Differential Equations on Surfaces: The Helmholtz-Beltrami Operator

We consider solving the surface Helmholtz equation on a smooth two dimensional surface embedded into a three dimensional space meshed with tetrahedra. The mesh does not respect the surface and thus the surface cuts through the elements. We consider a Galerkin method based on using the restrictions of continuous piecewise linears defined on the tetrahedra to the surface as trial and test functions.Using a stabilized method combining Galerkin least squares stabilization and a penalty on the gradient jumps we obtain stability of the discrete formulation under the condition $h k < C$, where $h$ denotes the mesh size, $k$ the wave number and $C$ a constant depending mainly on the surface curvature $κ$, but not on the surface/mesh intersection. Optimal error estimates in the $H^1$ and $L^2$-norms follow.

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arXiv2018

A Nitsche-based formulation for fluid-structure interactions with contact

We derive a Nitsche-based formulation for fluid-structure interaction (FSI) problems with contact. The approach is based on the work of Chouly and Hild [SIAM Journal on Numerical Analysis. 2013;51(2):1295--1307] for contact problems in solid mechanics. We present two numerical approaches, both of them formulating the FSI interface and the contact conditions simultaneously in equation form on a joint interface-contact surface $Γ(t)$. The first approach uses a relaxation of the contact conditions to allow for a small mesh-dependent gap between solid and wall. The second alternative introduces an artificial fluid below the contact surface. The resulting systems of equations can be included {in a consistent fashion} within a monolithic variational formulation, which prevents the so-called "chattering" phenomenon. To deal with the topology changes in the fluid domain at the time of impact, we use a fully Eulerian approach for the FSI problem. We compare the effect of slip and no-slip interface conditions and study the performance of the method by means of numerical examples.

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arXiv2018

A Stabilized Cut Streamline Diffusion Finite Element Method for Convection-Diffusion Problems on Surfaces

We develop a stabilized cut finite element method for the stationary convection diffusion problem on a surface embedded in ${\mathbb{R}}^d$. The cut finite element method is based on using an embedding of the surface into a three dimensional mesh consisting of tetrahedra and then using the restriction of the standard piecewise linear continuous elements to a piecewise linear approximation of the surface. The stabilization consists of a standard streamline diffusion stabilization term on the discrete surface and a so called normal gradient stabilization term on the full tetrahedral elements in the active mesh. We prove optimal order a priori error estimates in the standard norm associated with the streamline diffusion method and bounds for the condition number of the resulting stiffness matrix. The condition number is of optimal order $O(h^{-1})$ for a specific choice of method parameters. Numerical example supporting our theoretical results are also included.

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arXiv2018

Cut Finite Elements for Convection in Fractured Domains

We develop a cut finite element method (CutFEM) for the convection problem in a so called fractured domain which is a union of manifolds of different dimensions such that a $d$ dimensional component always resides on the boundary of a $d+1$ dimensional component. This type of domain can for instance be used to model porous media with embedded fractures that may intersect. The convection problem can be formulated in a compact form suitable for analysis using natural abstract directional derivative and divergence operators. The cut finite element method is based on using a fixed background mesh that covers the domain and the manifolds are allowed to cut through a fixed background mesh in an arbitrary way. We consider a simple method based on continuous piecewise linear elements together with weak enforcement of the coupling conditions and stabilization. We prove a priori error estimates and present illustrating numerical examples.

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arXiv2018

A Cut Finite Element Method for Elliptic Bulk Problems with Embedded Surfaces

We propose an unfitted finite element method for flow in fractured porous media. The coupling across the fracture uses a Nitsche type mortaring, allowing for an accurate representation of the jump in the normal component of the gradient of the discrete solution across the fracture. The flow field in the fracture is modelled simultaneously, using the average of traces of the bulk variables on the fractured. In particular the Laplace-Beltrami operator for the transport in the fracture is included using the average of the projection on the tangential plane of the fracture of the trace of the bulk gradient. Optimal order error estimates are proven under suitable regularity assumptions on the domain geometry. The extension to the case of bifurcating fractures is discussed. Finally the theory is illustrated by a series of numerical examples.

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arXiv2018

An unfitted Hybrid High-Order method for elliptic interface problems

We design and analyze a Hybrid High-Order (HHO) method on unfitted meshes to approximate elliptic interface problems. The curved interface can cut through the mesh cells in a very general fashion. As in classical HHO methods, the present unfitted method introduces cell and face unknowns in uncut cells, but doubles the unknowns in the cut cells and on the cut faces. The main difference with classical HHO methods is that a Nitsche-type formulation is used to devise the local reconstruction operator. As in classical HHO methods, cell unknowns can be eliminated locally leading to a global problem coupling only the face unknowns by means of a compact stencil. We prove stability estimates and optimal error estimates in the $H^1$-norm. Robustness with respect to cuts is achieved by a local cell-agglomeration procedure taking full advantage of the fact that HHO methods support polyhedral meshes. Robustness with respect to the contrast in the material properties from both sides of the interface is achieved by using material-dependent weights in Nitsche's formulation.

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arXiv2018

A cut finite element method with boundary value correction for the incompressible Stokes' equations

We design a cut finite element method for the incompressible Stokes equations on curved domains. The cut finite element method allows for the domain boundary to cut through the elements of the computational mesh in a very general fashion. To further facilitate the implementation we propose to use a piecewise affine discrete domain even if the physical domain has curved boundary. Dirichlet boundary conditions are imposed using Nitsche's method on the discrete boundary and the effect of the curved physical boundary is accounted for using the boundary value correction technique introduced for cut finite element methods in Burman, Hansbo, Larson, 'A cut finite element method with boundary value correction', Math. Comp. 87(310):633--657, 2018.

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arXiv2017

Primal dual mixed finite element methods for the elliptic Cauchy problem

We consider primal-dual mixed finite element methods for the solution of the elliptic Cauchy problem, or other related data assimilation problems. The method has a local conservation property. We derive a priori error estimates using known conditional stability estimates and determine the minimal amount of weakly consistent stabilization and Tikhonov regularization that yields optimal convergence for smooth exact solutions. The effect of perturbations in data is also accounted for. A reduced version of the method, obtained by choosing a special stabilization of the dual variable, can be viewed as a variant of the least squares mixed finite element method introduced by Dardé, Hannukainen and Hyvönen in \emph{An {$H\sb {\sf{div}}$}-based mixed quasi-reversibility method for solving elliptic {C}auchy problems}, SIAM J. Numer. Anal., 51(4) 2013. The main difference is that our choice of regularization does not depend on auxiliary parameters, the mesh size being the only asymptotic parameter. Finally, we show that the reduced method can be used for defect correction iteration to determine the solution of the full method. The theory is illustrated by some numerical examples.

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arXiv2017

A Simple Finite Element Method for Elliptic Bulk Problems with Embedded Surfaces

In this paper we develop a simple finite element method for simulation of embedded layers of high permeability in a matrix of lower permeability using a basic model of Darcy flow in embedded cracks. The cracks are allowed to cut through the mesh in arbitrary fashion and we take the flow in the crack into account by superposition. The fact that we use continuous elements leads to suboptimal convergence due to the loss of regularity across the crack. We therefore refine the mesh in the vicinity of the crack in order to recover optimal order convergence in terms of the global mesh parameter. The proper degree of refinement is determined based on an a priori error estimate and can thus be performed before the actual finite element computation is started. Numerical examples showing this effect and confirming the theoretical results are provided. The approach is easy to implement and beneficial for rapid assessment of the effect of crack orientation and may for example be used in an optimization loop.

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arXiv2017

Fully discrete finite element data assimilation method for the heat equation

We consider a finite element discretization for the reconstruction of the final state of the heat equation, when the initial data is unknown, but additional data is given in a sub domain in the space time. For the discretization in space we consider standard continuous affine finite element approximation, and the time derivative is discretized using a backward differentiation. We regularize the discrete system by adding a penalty of the $H^1$-semi-norm of the initial data, scaled with the mesh-parameter. The analysis of the method uses techniques developed in E. Burman and L. Oksanen, Data assimilation for the heat equation using stabilized finite element methods, arXiv, 2016, combining discrete stability of the numerical method with sharp Carleman estimates for the physical problem, to derive optimal error estimates for the approximate solution. For the natural space time energy norm, away from $t=0$, the convergence is the same as for the classical problem with known initial data, but contrary to the classical case, we do not obtain faster convergence for the $L^2$-norm at the final time.

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