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Erik Burman

Publications and source records attributed to Erik Burman.

124 records · Page 7Linked to original sources
arXiv2015

A penalty free Nitsche type method for the weak imposition of boundary conditions in compressible and incompressible elasticity

In this paper, we study the stability of the nonsymmetric version of Nitsche's method without penalty for compressible and incompressible elasticity. For the compressible case we prove the convergence of the error in the $H^1$- and $L^2$-norms. In the incompressible case we use a Galerkin least squares pressure stabilization and we prove the convergence in the $H^1$-norm for the velocity and convergence of the pressure in the $L^2$-norm.

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arXiv2015

A Cut Discontinuous Galerkin Method for the Laplace-Beltrami Operator

We develop a discontinuous cut finite element method (CutFEM) for the Laplace-Beltrami operator on a hypersurface embedded in $\mathbb{R}^d$. The method is constructed by using a discontinuous piecewise linear finite element space defined on a background mesh in $\mathbb{R}^d$. The surface is approximated by a continuous piecewise linear surface that cuts through the background mesh in an arbitrary fashion. Then a discontinuous Galerkin method is formulated on the discrete surface and in order to obtain coercivity, certain stabilization terms are added on the faces between neighboring elements that provide control of the discontinuity as well as the jump in the gradient. We derive optimal a priori error and condition number estimates which are independent of the positioning of the surface in the background mesh. Finally, we present numerical examples confirming our theoretical results.

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arXiv2015

A Cut Finite Element Method with Boundary Value Correction

In this contribution we develop a cut finite element method with boundary value correction of the type originally proposed by Bramble, Dupont, and Thomee. The cut finite element method is a fictitious domain method with Nitsche type enforcement of Dirichlet conditions together with stabilization of the elements at the boundary which is stable and enjoy optimal order approximation properties. A computational difficulty is, however, the geometric computations related to quadrature on the cut elements which must be accurate enough to achieve higher order approximation. With boundary value correction we may use only a piecewise linear approximation of the boundary, which is very convenient in a cut finite element method, and still obtain optimal order convergence. The boundary value correction is a modified Nitsche formulation involving a Taylor expansion in the normal direction compensating for the approximation of the boundary. Key to the analysis is a consistent stabilization term which enables us to prove stability of the method and a priori error estimates with explicit dependence on the meshsize and distance between the exact and approximate boundary.

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arXiv2015

A Stabilized Cut Finite Element Method for the Three Field Stokes Problem

We propose a Nitsche-based fictitious domain method for the three field Stokes problem in which the boundary of the domain is allowed to cross through the elements of a fixed background mesh. The dependent variables of velocity, pressure and extra-stress tensor are discretised on the background mesh using linear finite elements. This equal order approximation is stabilized using a continuous interior penalty (CIP) method. On the unfitted domain boundary, Dirichlet boundary conditions are weakly enforced using Nitsche's method. We add CIP-like ghost penalties in the boundary region and prove that our scheme is inf-sup stable and that it has optimal convergence properties independent of how the domain boundary intersects the mesh. Additionally, we demonstrate that the condition number of the system matrix is bounded independently of the boundary location. We corroborate our theoretical findings with numerical examples.

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arXiv2014

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 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. The resulting discrete method may be severely ill-conditioned, and the main purpose of this paper is to suggest a remedy for this problem based on adding a consistent stabilization term to the original bilinear form. We show optimal estimates for the condition number of the stabilized method independent of the location of the surface. We also prove optimal a priori error estimates for the stabilized method.

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arXiv2014

Error estimates for stabilized finite element methods applied to ill-posed problems

We propose an analysis for the stabilized finite element methods proposed in, E. Burman, Stabilized finite element methods for nonsymmetric, noncoercive, and ill-posed problems. Part I: Elliptic equations. SIAM J. Sci. Comput., 35(6) 2013, valid in the case of ill-posed problems for which only weak continuous dependence can be assumed. A priori and a posteriori error estimates are obtained without assuming coercivity or inf-sup stability of the continuous problem. A numerical example illustrates the theory.

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arXiv2014

A stabilized nonconforming finite element method for the elliptic Cauchy problem

In this paper we propose a nonconforming finite element method for the solution of the ill-posed elliptic Cauchy problem. We prove error estimates using continuous dependence estimates in the $L^2$-norm. The effect of perturbations in data on the estimates is investigated. The recently derived framework from \cite{Bu13,Bu14} is extended to include the case of nonconforming approximation spaces and we show that the use of such spaces allows us to reduce the amount of stabilization necessary for convergence, even in the case of ill-posed problems.

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arXiv2014

Error estimates for shock capturing finite element approximations of the one dimensional Burgers' equation

We propose an error analysis in weak norms of a shock capturing finite element method for the Burgers' equation. The estimates can be related to estimates of certain filtered quantities and are robust in the inviscid limit. Using a total variation apriori bound on the discrete solution and an interpolation inequality error estimates in $L^p$-norms can are obtained using interpolation.

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arXiv2014

Robust error estimates in weak norms for advection dominated transport problems with rough data

We consider mixing problems in the form of transient convection--diffusion equations with a velocity vector field with multiscale character and rough data. We assume that the velocity field has two scales, a coarse scale with slow spatial variation, which is responsible for advective transport and a fine scale with small amplitude that contributes to the mixing. For this problem we consider the estimation of filtered error quantities for solutions computed using a finite element method with symmetric stabilization. A posteriori error estimates and a priori error estimates are derived using the multiscale decomposition of the advective velocity to improve stability. All estimates are independent both of the Péclet number and of the regularity of the exact solution.

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arXiv2014

Stabilized finite element methods for nonsymmetric, noncoercive and ill-posed problem. Part II: hyperbolic equations

In this paper we consider stabilised finite element methods for hyperbolic transport equations without coercivity. Abstract conditions for the convergence of the methods are introduced and these conditions are shown to hold for three different stabilised methods: the Galerkin least squares method, the continuous interior penalty method and the discontinuous Galerkin method. We consider both the standard stabilisation methods and the optimisation based method introduced in \cite{part1}. The main idea of the latter is to write the stabilised method in an optimisation framework and select the discrete function for which a certain cost functional, in our case stabilisation term, is minimised. Some numerical examples illustrate the theoretical investigations.

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arXiv2014

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 $L^2$ 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.

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arXiv2013

Projection stabilisation of Lagrange multipliers for the imposition of constraints on interfaces and boundaries

Projection stabilisation applied to general Lagrange multiplier finite element methods is introduced and analysed in an abstract framework. We then consider some applications of the stabilised methods: (i) the weak imposition of boundary conditions, (ii) multiphysics coupling on unfitted meshes, (iii) a new interpretation of the classical residual stabilised Lagrange multiplier method introduced in H. J. C. Barbosa and T. J. R. Hughes, The finite element method with Lagrange multipliers on the boundary: circumventing the Babuška-Brezzi condition, Comput. Methods Appl. Mech. Engrg., 85(1):109--128, 1991 .

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arXiv2013

Stabilised finite element methods for non-symmetric, non-coercive and ill-posed problems. Part I: elliptic equations

In this paper we propose a new method to stabilise non-symmetric indefinite problems. The idea is to solve a forward and an adjoint problem simultaneously using a suitable stabilised finite element method. Both stabilisation of the element residual and jumps of certain derivatives of the discrete solution over element faces may be used. Under the assumption of well posedness of the partial differential equation and its associated adjoint problem we prove optimal error estimates in $H^1$ and $L^2$ norms in an abstract framework. Some examples of problems that are neither symmetric nor coercive, but that enter the abstract framework are given. First we treat indefinite convection-diffusion equations, with non-solenoidal transport velocity and either pure Dirichlet conditions or pure Neumann conditions and then a Cauchy problem for the Helmholtz operator. Some numerical illustrations are given.

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arXiv2013

Robust error estimates for stabilized finite element approximations of the two dimensional Navier-Stokes equations with application to implicit large eddy simulation

We consider error estimates in weak parametrised norms for stabilized finite element approximations of the two-dimensional Navier-Stokes' equations. These weak norms can be related to the norms of certain filtered quantities, where the parameter of the norm, relates to the filter width. Under the assumption of the existence of a certain decomposition of the solution, into large eddies and fine scale fluctuations, the constants of the estimates are proven to be independent of both the Reynolds number and the Sobolev norm of the exact solution. Instead they exhibit exponential growth with a coefficient proportional to the maximum gradient of the large eddies. The error estimates are on a posteriori form, but using Sobolev injections valid on finite element spaces and the properties of the stabilization operators the residuals may be upper bounded uniformly, leading to robust a priori error estimates.

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arXiv2012

Continuous Interior Penalty Finite Element Method for Helmholtz Equation with High Wave Number: One Dimensional Analysis

This paper addresses the properties of Continuous Interior Penalty (CIP) finite element solutions for the Helmholtz equation. The $h$-version of the CIP finite element method with piecewise linear approximation is applied to a one-dimensional model problem. We first show discrete well posedness and convergence results, using the imaginary part of the stabilization operator, for the complex Helmholtz equation. Then we consider a method with real valued penalty parameter and prove an error estimate of the discrete solution in the $H^1$-norm, as the sum of best approximation plus a pollution term that is the order of the phase difference. It is proved that the pollution can be eliminated by selecting the penalty parameter appropriately. As a result of this analysis, thorough and rigorous understanding of the error behavior throughout the range of convergence is gained. Numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior.

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