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Endre Suli - One of the best experts on this subject based on the ideXlab platform.
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Finite Element Approximation of steady flows of colloidal solutions
'EDP Sciences', 2021Co-Authors: Andrea Bonito, Vivette Girault, Diane Guignard, Kumbakonam R. Rajagopal, Endre SuliAbstract:We consider the mathematical analysis and numerical Approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We prove the existence of a weak solution to the problem, and under the additional assumption that the nonlinearity involved in the constitutive relation is Lipschitz continuous we also prove uniqueness of the weak solution. We then construct mixed Finite Element Approximations of the system using both conforming and nonconforming Finite Element spaces. For both of these we prove the convergence of the method to the unique weak solution of the problem, and in the case of the conforming method we provide a bound on the error between the analytical solution and its Finite Element Approximation in terms of the best Approximation error from the Finite Element spaces. We propose first a Lions–Mercier type iterative method and next a classical fixed-point algorithm to solve the Finite-dimensional problems resulting from the Finite Element discretisation of the system of nonlinear partial differential equations under consideration and present numerical experiments that illustrate the practical performance of the proposed numerical method
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fully discrete Finite Element Approximation of unsteady flows of implicitly constituted incompressible fluids
Ima Journal of Numerical Analysis, 2020Co-Authors: Endre Suli, Tabea TscherpelAbstract:Implicit constitutive theory provides a very general framework for fluid flow models, including both Newtonian and generalized Newtonian fluids, where the Cauchy stress tensor and the rate of strain tensor are assumed to be related by an implicit relation associated with a maximal monotone graph. For incompressible unsteady flows of such fluids, subject to a homogeneous Dirichlet boundary condition on a Lipschitz polytopal domain $\Omega \subset \mathbb{R}^d$, $d \in \{2,3\}$, we investigate a fully-discrete Approximation scheme, using a spatial mixed Finite Element Approximation combined with backward Euler time-stepping. We show convergence of a subsequence of approximate solutions, when the velocity field belongs to the space of solenoidal functions contained in $L^\infty(0,T;L^2(\Omega)^d)\cap L^q(0,T;W^{1,q}_0(\Omega)^d)$, provided that $q\in \big(\frac{2d}{d+2},\infty\big)$, which is the maximal range for $q$ with respect to existence of weak solutions. This is achieved by a technique based on splitting and regularizing, the use of a solenoidal parabolic Lipschitz truncation method, a local Minty-type monotonicity result, and various weak compactness results.
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Finite Element Approximation of elliptic homogenization problems in nondivergence-form
'EDP Sciences', 2020Co-Authors: Yves Capdeboscq, Timo Sprekeler, Endre SuliAbstract:We use uniform W2,p estimates to obtain corrector results for periodic homogenization problems of the form A(x/ε):D2uε = f subject to a homogeneous Dirichlet boundary condition. We propose and rigorously analyze a numerical scheme based on Finite Element Approximations for such nondivergence-form homogenization problems. The second part of the paper focuses on the Approximation of the corrector and numerical homogenization for the case of nonuniformly oscillating coefficients. Numerical experiments demonstrate the performance of the scheme
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mixed Finite Element Approximation of the hamilton jacobi bellman equation with cordes coefficients
SIAM Journal on Numerical Analysis, 2019Co-Authors: Dietmar Gallistl, Endre SuliAbstract:A mixed Finite Element Approximation of $H^2$ solutions to the fully nonlinear Hamilton--Jacobi--Bellman equation, with coefficients that satisfy the Cordes condition, is proposed and analyzed. A p...
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Finite Element Approximation of a strain limiting elastic model
arXiv: Numerical Analysis, 2018Co-Authors: Andrea Bonito, Vivette Girault, Endre SuliAbstract:We construct a Finite Element Approximation of a strain-limiting elastic model on a bounded open domain in $\mathbb{R}^d$, $d \in \{2,3\}$. The sequence of Finite Element Approximations is shown to exhibit strong convergence to the unique weak solution of the model. Assuming that the material parameters featuring in the model are Lipschitz-continuous, and assuming that the weak solution has additional regularity, the sequence of Finite Element Approximations is shown to converge with a rate. An iterative algorithm is constructed for the solution of the system of nonlinear algebraic equations that arises from the Finite Element Approximation. An appealing feature of the iterative algorithm is that it decouples the monotone and linear elastic parts of the nonlinearity in the model. In particular, our choice of piecewise constant Approximation for the stress tensor (and continuous piecewise linear Approximation for the displacement) allows us to compute the monotone part of the nonlinearity by solving an algebraic system with $d(d+1)/2$ unknowns independently on each Element in the subdivision of the computational domain. The theoretical results are illustrated by numerical experiments.
John W. Barrett - One of the best experts on this subject based on the ideXlab platform.
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Finite Element Approximation of the fene p model
arXiv: Numerical Analysis, 2017Co-Authors: John W. Barrett, Sebastien BoyavalAbstract:We extend our analysis on the Oldroyd-B model in Barrett and Boyaval [1] to consider the Finite Element Approximation of the FENE-P system of equations, which models a dilute polymeric fluid, in a bounded domain $D $\subset$ R d , d = 2 or 3$, subject to no flow boundary conditions. Our schemes are based on approximating the pressure and the symmetric conforma-tion tensor by either (a) piecewise constants or (b) continuous piecewise linears. In case (a) the velocity field is approximated by continuous piecewise quadratics ($d = 2$) or a reduced version, where the tangential component on each simplicial edge ($d = 2$) or face ($d = 3$) is linear. In case (b) the velocity field is approximated by continuous piecewise quadratics or the mini-Element. We show that both of these types of schemes, based on the backward Euler type time discretiza-tion, satisfy a free energy bound, which involves the logarithm of both the conformation tensor and a linear function of its trace, without any constraint on the time step. Furthermore, for our Approximation (b) in the presence of an additional dissipative term in the stress equation, the so-called FENE-P model with stress diffusion, we show (subsequence) convergence in the case $d = 2$, as the spatial and temporal discretization parameters tend to zero, towards global-in-time weak solutions of this FENE-P system. Hence, we prove existence of global-in-time weak solutions to the FENE-P model with stress diffusion in two spatial dimensions.
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Numerische Mathematik manuscript No. (will be inserted by the editor) Finite Element Approximation of a Nonlinear Cross-Diffusion Population Model
2015Co-Authors: John W. Barrett, James F BloweyAbstract:Summary We consider a fully discrete Finite Element Approximation of the nonlinear cross-diffusion population model: Find u i, the population of the ith species, i = 1 and 2, such that ∂ui ∂t −Δ [ ci ui + ai u2i + ui uj] − bi ∇. (ui∇v) = gi(u1, u2), where j = i and gi(u1, u2): = (μi − γii ui − γij uj) ui. In the above, the given data is as follows: v is an environmental potential, c i ∈ R≥0, ai ∈ R>0 are diffusion coefficients, bi ∈ R are transport coefficients, μi ∈ R≥0 are the intrinsic growth rates, and γii ∈ R≥0 are intra-specific, whereas γij, i = j, ∈ R≥0 are interspecific competition coefficients. In addition to showing well-posedness of our Approximation, we prove convergence in space dimensions d ≤ 3. Finally some numerical experiments in one space dimension are presented. Mathematics Subject Classification (1991):65M60, 65M12, 35K55, 92D25
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Finite Element Approximation of coupled surface and grain boundary motion with applications to thermal grooving and sintering
European Journal of Applied Mathematics, 2010Co-Authors: John W. Barrett, Harald Garcke, Robert NurnbergAbstract:We study the coupled surface and grain boundary motion in bi- and tricrystals in three-space dimensions, building on previous work by the authors on the simplified two-dimensional case. The motion of the interfaces, which in this paper are presented by two-dimensional hypersurfaces, is described by two types of normal velocities: motion by mean curvature and motion by surface diffusion. Three hypersurfaces meet at triple-junction lines, where junction conditions need to hold. Similarly, boundary conditions are prescribed where an interface meets an external boundary, and these conditions naturally give rise to contact angles. We present a variational formulation of the flows, which leads to a fully practical Finite-Element Approximation that exhibits excellent mesh properties, with no mesh smoothing or remeshing required in practice. For the introduced parametric Finite-Element Approximation we show well posedness and, in general, unconditional stability, i.e. there is no restriction on the chosen time-step size. Moreover, the induced discrete equations are linear and easy to solve. A generalisation to anisotropic surface energies is straightforward. Several numerical results in two- and three-space dimensions are presented, including simulations for thermal grooving and sintering. Three-dimensional simulations featuring quadruple junction points, non-standard boundary contact angles and fully anisotropic surface energies are also presented.
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Finite Element Approximation of a nonlinear cross diffusion population model
Numerische Mathematik, 2004Co-Authors: John W. Barrett, James F BloweyAbstract:We consider a fully discrete Finite Element Approximation of the nonlinear cross-diffusion population model: Find ui, the population of the ith species, i=1 and 2, such that * where j≠i and gi(u1,u2):=(μi−γii ui−γij uj) ui. In the above, the given data is as follows: v is an environmental potential, ci ∈ ℝ, ai ∈ ℝ are diffusion coefficients, bi ∈ ℝ are transport coefficients, μi ∈ ℝ are the intrinsic growth rates, and γii ∈ ℝ are intra-specific, whereas γij, i≠j, ∈ ℝ are interspecific competition coefficients. In addition to showing well-posedness of our Approximation, we prove convergence in space dimensions d≤3. Finally some numerical experiments in one space dimension are presented.
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Finite Element Approximation of a sixth order nonlinear degenerate parabolic equation
Numerische Mathematik, 2004Co-Authors: John W. Barrett, Stephen Langdon, Robert NurnbergAbstract:We consider a Finite Element Approximation of the sixth order nonlinear degenerate parabolic equation **** equation here *** where generically **** equation here *** for any given **** equation here *** In addition to showing well-posedness of our Approximation, we prove convergence in space dimensions $d \leq 3$. Furthermore an iterative scheme for solving the resulting nonlinear discrete system is analysed. Finally some numerical experiments in one and two space dimensions are presented.
Ramon Codina - One of the best experts on this subject based on the ideXlab platform.
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Residual-based stabilization of the Finite Element Approximation to the acoustic perturbation equations for low Mach number aeroacoustics
International Journal for Numerical Methods in Fluids, 2016Co-Authors: Oriol Guasch, Patricia Sánchez-martín, Arnau Pont, Joan Baiges, Ramon CodinaAbstract:This is the peer reviewed version of the following article: [Guasch, O., Sanchez-Martin, P., Pont, A., Baiges, J., and Codina, R. (2016) Residual-based stabilization of the Finite Element Approximation to the acoustic perturbation equations for low Mach number aeroacoustics. Int. J. Numer. Meth. Fluids, 82: 839–857. doi: 10.1002/fld.4243], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/fld.4243/abstract. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.
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explicit reduced order models for the stabilized Finite Element Approximation of the incompressible navier stokes equations
International Journal for Numerical Methods in Fluids, 2013Co-Authors: Joan Baiges, Ramon Codina, Sergio IdelsohnAbstract:SUMMARY In this paper, we present an explicit formulation for reduced-order models of the stabilized Finite Element Approximation of the incompressible Navier–Stokes equations. The basic idea is to build a reduced-order model based on a proper orthogonal decomposition and a Galerkin projection and treat all the terms in an explicit way in the time integration scheme, including the pressure. This is possible because the reduced model snapshots do already fulfill the continuity equation. The pressure field is automatically recovered from the reduced-order basis and solution coefficients. The main advantage of this explicit treatment of the incompressible Navier–Stokes equations is that it allows for the easy use of hyper-reduced order models, because only the right-hand side vector needs to be recovered by means of a gappy data reconstruction procedure. A method for choosing the optimal set of sampling points at the discrete level in the gappy procedure is also presented. Numerical examples show the performance of the proposed strategy. Copyright © 2013 John Wiley & Sons, Ltd.
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a nodal based Finite Element Approximation of the maxwell problem suitable for singular solutions
SIAM Journal on Numerical Analysis, 2012Co-Authors: Santiago Badia, Ramon CodinaAbstract:A new mixed Finite Element Approximation of Maxwell's problem is proposed, its main features being that it is based on a novel augmented formulation of the continuous problem and the introduction of a mesh dependent stabilizing term, which yields a very weak control on the divergence of the unknown. The method is shown to be stable and convergent in the natural $H({\rm \mathbf{curl}}\, 0; \Omega)$ norm for this unknown. In particular, convergence also applies to singular solutions, for which classical nodal-based interpolations are known to suffer from spurious convergence upon mesh refinement.
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Finite Element Approximation of transmission conditions in fluids and solids introducing boundary subgrid scales
International Journal for Numerical Methods in Engineering, 2011Co-Authors: Ramon Codina, Joan BaigesAbstract:Terms involving jumps of stresses on boundaries are proposed for the Finite Element Approximation of the Stokes problem and the linear elasticity equations. These terms are designed to improve the transmission conditions between subdomains at three different levels, namely, between the Element domains, between the interfaces in homogeneous domain interaction problems and at the interface between the fluid and the solid in fluid–structure interaction problems. The benefits in each case are respectively the possibility of using discontinuous pressure interpolations in a stabilized Finite Element Approximation of the Stokes problem, a stronger enforcement of the stress continuity in homogeneous domain decomposition problems and a considerable improvement of the behavior of iterative schemes to couple the fluid and the solid in fluid–structure integration algorithms. The motivation to introduce these terms stems from a decomposition of the unknown into a conforming and a non-conforming part, a hybrid formulation for the latter and a simple Approximation for the unknowns involved in the hybrid problem. Copyright © 2011 John Wiley & Sons, Ltd.
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Finite Element Approximation of turbulent thermally coupled incompressible flows with numerical sub grid scale modeling
International Journal of Numerical Methods for Heat & Fluid Flow, 2010Co-Authors: Ramon Codina, Javier Principe, Matias AvilaAbstract:Purpose – The purpose of this paper is to describe a variational multiscale Finite Element Approximation for the incompressible Navier‐Stokes equations using the Boussinesq Approximation to model thermal coupling.Design/methodology/approach – The main feature of the formulation, in contrast to other stabilized methods, is that the subscales are considered as transient and orthogonal to the Finite Element space. These subscales are solution of a differential equation in time that needs to be integrated. Likewise, the effect of the subscales is kept, both in the nonlinear convective terms of the momentum and temperature equations and, if required, in the thermal coupling term of the momentum equation.Findings – This strategy allows the approaching of the problem of dealing with thermal turbulence from a strictly numerical point of view and discussion important issues, such as the relationship between the turbulent mechanical dissipation and the turbulent thermal dissipation.Originality/value – The treatment...
James F Blowey - One of the best experts on this subject based on the ideXlab platform.
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Numerische Mathematik manuscript No. (will be inserted by the editor) Finite Element Approximation of a Nonlinear Cross-Diffusion Population Model
2015Co-Authors: John W. Barrett, James F BloweyAbstract:Summary We consider a fully discrete Finite Element Approximation of the nonlinear cross-diffusion population model: Find u i, the population of the ith species, i = 1 and 2, such that ∂ui ∂t −Δ [ ci ui + ai u2i + ui uj] − bi ∇. (ui∇v) = gi(u1, u2), where j = i and gi(u1, u2): = (μi − γii ui − γij uj) ui. In the above, the given data is as follows: v is an environmental potential, c i ∈ R≥0, ai ∈ R>0 are diffusion coefficients, bi ∈ R are transport coefficients, μi ∈ R≥0 are the intrinsic growth rates, and γii ∈ R≥0 are intra-specific, whereas γij, i = j, ∈ R≥0 are interspecific competition coefficients. In addition to showing well-posedness of our Approximation, we prove convergence in space dimensions d ≤ 3. Finally some numerical experiments in one space dimension are presented. Mathematics Subject Classification (1991):65M60, 65M12, 35K55, 92D25
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Finite Element Approximation of a nonlinear cross diffusion population model
Numerische Mathematik, 2004Co-Authors: John W. Barrett, James F BloweyAbstract:We consider a fully discrete Finite Element Approximation of the nonlinear cross-diffusion population model: Find ui, the population of the ith species, i=1 and 2, such that * where j≠i and gi(u1,u2):=(μi−γii ui−γij uj) ui. In the above, the given data is as follows: v is an environmental potential, ci ∈ ℝ, ai ∈ ℝ are diffusion coefficients, bi ∈ ℝ are transport coefficients, μi ∈ ℝ are the intrinsic growth rates, and γii ∈ ℝ are intra-specific, whereas γij, i≠j, ∈ ℝ are interspecific competition coefficients. In addition to showing well-posedness of our Approximation, we prove convergence in space dimensions d≤3. Finally some numerical experiments in one space dimension are presented.
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Finite Element Approximation of the cahn hilliard equation with degenerate mobility
SIAM Journal on Numerical Analysis, 1999Co-Authors: John W. Barrett, James F Blowey, Harald GarckeAbstract:We consider a fully practical Finite Element Approximation of the Cahn--Hilliard equation with degenerate mobility $$ \textstyle \frac{\partial u}{\partial t}= \del .(\,b(u)\, \del (-\gamma\lap u+\Psi'(u))) , $$ where $b(\cdot)\geq 0$ is a diffusional mobility and $\Psi(\cdot)$ is a homogeneous free energy. In addition to showing well posedness and stability bounds for our Approximation, we prove convergence in one space dimension. Furthermore, an iterative scheme for solving the resulting nonlinear discrete system is analyzed. We also discuss how our Approximation has to be modified in order to be applicable to a logarithmic homogeneous free energy. Finally, some numerical experiments are presented.
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Finite Element Approximation of the cahn hilliard equation with concentration dependent mobility
Mathematics of Computation, 1999Co-Authors: John W. Barrett, James F BloweyAbstract:We consider the Cahn-Hilliard equation with a logarithmic free energy and non-degenerate concentration dependent mobility. In particular we prove that there exists a unique solution for sufficiently smooth initial data. Further, we prove an error bound for a fully practical piecewise linear Finite Element Approximation in one and two space dimensions. Finally some numerical experiments are presented.
Weizhang Huang - One of the best experts on this subject based on the ideXlab platform.
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Stability of Explicit One-Step Methods for P1-Finite Element Approximation of Linear Diffusion Equations on Anisotropic Meshes
SIAM Journal on Numerical Analysis, 2016Co-Authors: Weizhang Huang, Lennard Kamenski, Jens LangAbstract:We study the stability of explicit one-step integration schemes for the linear Finite Element Approximation of linear parabolic equations. The derived bound on the largest permissible time step is tight for any mesh and any diffusion matrix within a factor of $2(d+1)$, where $d$ is the spatial dimension. Both full mass matrix and mass lumping are considered. The bound reveals that the stability condition is affected by two factors. The first depends on the number of mesh Elements and corresponds to the classic bound for the Laplace operator on a uniform mesh. The second factor reflects the effects of the interplay of the mesh geometry and the diffusion matrix. It is shown that it is not the mesh geometry itself but the mesh geometry in relation to the diffusion matrix that is crucial to the stability of explicit methods. When the mesh is uniform in the metric specified by the inverse of the diffusion matrix, the stability condition is comparable to the situation with the Laplace operator on a uniform mesh...
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stability of explicit one step methods for p1 Finite Element Approximation of linear diffusion equations on anisotropic meshes
arXiv: Numerical Analysis, 2016Co-Authors: Weizhang Huang, Lennard Kamenski, Jens LangAbstract:We study the stability of explicit one-step integration schemes for the linear Finite Element Approximation of linear parabolic equations. The derived bound on the largest permissible time step is tight for any mesh and any diffusion matrix within a factor of $2(d+1)$, where $d$ is the spatial dimension. Both full mass matrix and mass lumping are considered. The bound reveals that the stability condition is affected by two factors. The first one depends on the number of mesh Elements and corresponds to the classic bound for the Laplace operator on a uniform mesh. The other factor reflects the effects of the interplay of the mesh geometry and the diffusion matrix. It is shown that it is not the mesh geometry itself but the mesh geometry in relation to the diffusion matrix that is crucial to the stability of explicit methods. When the mesh is uniform in the metric specified by the inverse of the diffusion matrix, the stability condition is comparable to the situation with the Laplace operator on a uniform mesh. Numerical results are presented to verify the theoretical findings.
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Stability of Explicit Runge-Kutta Methods for High Order Finite Element Approximation of Linear Parabolic Equations
Lecture Notes in Computational Science and Engineering, 2014Co-Authors: Weizhang Huang, Lennard Kamenski, Jens LangAbstract:We study the stability of explicit Runge-Kutta methods for high order Lagrangian Finite Element Approximation of linear parabolic equations and establish bounds on the largest eigenvalue of the system matrix which determines the largest permissible time step. A bound expressed in terms of the ratio of the diagonal entries of the stiffness and mass matrices is shown to be tight within a small factor which depends only on the dimension and the choice of the reference Element and basis functions but is independent of the mesh or the coefficients of the initial-boundary value problem under consideration. Another bound, which is less tight and expressed in terms of mesh geometry, depends only on the number of mesh Elements and the alignment of the mesh with the diffusion matrix. The results provide an insight into how the interplay between the mesh geometry and the diffusion matrix affects the stability of explicit integration schemes when applied to a high order Finite Element Approximation of linear parabolic equations on general nonuniform meshes.
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sign preserving of principal eigenfunctions in p1 Finite Element Approximation of eigenvalue problems of second order elliptic operators
Journal of Computational Physics, 2014Co-Authors: Weizhang HuangAbstract:This paper is concerned with the P1 Finite Element Approximation of the eigenvalue problem of second-order elliptic operators subject to the Dirichlet boundary condition. The focus is on the preservation of the basic properties of the principal eigenvalue and eigenfunctions of the continuous problem. It is shown that when the stiffness matrix is an irreducible M-matrix, the discrete eigenvalue problem maintains almost all of the basic properties such as the smallest eigenvalue being real and simple and the corresponding eigenfunctions being either positive or negative inside the physical domain. Mesh conditions leading to such a stiffness matrix are also studied. A sufficient condition is that the mesh is simplicial, interiorly connected, and acute when measured in the metric specified by the inverse of the diffusion matrix. The acute requirement can be replaced by the Delaunay condition in two dimensions. Numerical results show that when the stiffness matrix is not an M-matrix, a Finite Element Approximation can be structurally different from the continuous eigenvalue problem: the eigenfunctions corresponding to the smallest eigenvalue can change sign over the physical domain and the smallest eigenvalue (in modulus) can even be complex for the case with nonsymmetric operators.