scholarly journals Adaptive Goal-Oriented Solver for the Linearized Poisson- Boltzmann Equation

2021 ◽  
Author(s):  
Svetoslav Nakov ◽  
Ekaterina Sobakinskaya ◽  
Thomas Renger ◽  
Johannes Kraus
Keyword(s):  
Finite Element ◽  
Sobolev Spaces ◽  
Numerical Experiments ◽  
Posteriori Error ◽  
Posteriori Error Estimate ◽  
Adaptive Finite Element ◽  
Original Goal ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation ◽  
Electrostatic Coupling

An adaptive finite element solver for the numerical calculation of the electrostatic coupling between molecules in a solvent environment is developed and tested. The new solver is based on a derivation of a new goal-oriented a posteriori error estimate for the electrostatic coupling. This estimate involves the consideration of the primal and adjoint problems for the electrostatic potential of the system, where the goal functional requires pointwise evaluations of the potential. A common practice to treat such functionals is to regularize them, for example, by averaging over balls or by mollification, and to use weak formulations in standard Sobolev spaces. However, this procedure changes the goal functional and may require the numerical evaluation of integrals of discontinuous functions. We overcome the conceptual shortcomings of this approach by using weak formulations involving nonstandard Sobolev spaces and deriving a representation of the error in the goal quantity which does not require averaging and directly exploits the original goal functional. The accuracy of this solver is evaluated by numerical experiments on a series of problems with analytically known solutions. In addition, the solver is used to calculate electrostatic couplings between two chromophores, linked to polyproline helices of different lengths. All the numerical experiments are repeated by using the well-known finite difference solvers MEAD and APBS, revealing the advantages of the present finite element solver.<br><br>

scholarly journals Adaptive Goal-Oriented Solver for the Linearized Poisson- Boltzmann Equation

2021 ◽  
Author(s):  
Svetoslav Nakov ◽  
Ekaterina Sobakinskaya ◽  
Thomas Renger ◽  
Johannes Kraus
Keyword(s):  
Finite Element ◽  
Sobolev Spaces ◽  
Numerical Experiments ◽  
Posteriori Error ◽  
Posteriori Error Estimate ◽  
Adaptive Finite Element ◽  
Original Goal ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation ◽  
Electrostatic Coupling

An adaptive finite element solver for the numerical calculation of the electrostatic coupling between molecules in a solvent environment is developed and tested. The new solver is based on a derivation of a new goal-oriented a posteriori error estimate for the electrostatic coupling. This estimate involves the consideration of the primal and adjoint problems for the electrostatic potential of the system, where the goal functional requires pointwise evaluations of the potential. A common practice to treat such functionals is to regularize them, for example, by averaging over balls or by mollification, and to use weak formulations in standard Sobolev spaces. However, this procedure changes the goal functional and may require the numerical evaluation of integrals of discontinuous functions. We overcome the conceptual shortcomings of this approach by using weak formulations involving nonstandard Sobolev spaces and deriving a representation of the error in the goal quantity which does not require averaging and directly exploits the original goal functional. The accuracy of this solver is evaluated by numerical experiments on a series of problems with analytically known solutions. In addition, the solver is used to calculate electrostatic couplings between two chromophores, linked to polyproline helices of different lengths. All the numerical experiments are repeated by using the well-known finite difference solvers MEAD and APBS, revealing the advantages of the present finite element solver.<br><br>

scholarly journals Adaptive Goal-Oriented Solver for the Linearized Poisson- Boltzmann Equation

2020 ◽  
Author(s):  
Svetoslav Nakov ◽  
Ekaterina Sobakinskaya ◽  
Thomas Renger ◽  
Johannes Kraus
Keyword(s):  
Finite Element ◽  
Numerical Experiments ◽  
A Posteriori Error Estimates ◽  
Posteriori Error ◽  
Adaptive Finite Element ◽  
A Posteriori ◽  
A Posteriori Error ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation ◽  
Electrostatic Coupling

An adaptive finite element solver for the numerical calculation of the electrostatic coupling between molecules is developed and verified. The development is based on a derivation of a goal-oriented a posteriori error estimates for the electrostatic coupling. These estimates involve the consideration of the primal and adjoint problems for the electrostatic potential of the system. The accuracy of this solver is evaluated by numerical experiments on a series of problems with analytically known solutions. In addition, the solver is used to calculate electrostatic couplings between two chromophores, linked to polyproline helices of different lengths. All the numerical experiments are repeated by using the well known finite difference solvers MEAD and APBS, revealing the advantages of the present finite element solver<br>

scholarly journals Adaptive modelling of variably saturated seepage problems

2021 ◽  
Author(s):  
B Ashby ◽  
C Bortolozo ◽  
A Lukyanov ◽  
T Pryer
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Degrees Of Freedom ◽  
A Priori ◽  
Water Flux ◽  
Posteriori Error ◽  
Adaptive Finite Element Method ◽  
Posteriori Error Estimate ◽  
Adaptive Finite Element ◽  
Element Method

Summary In this article, we present a goal-oriented adaptive finite element method for a class of subsurface flow problems in porous media, which exhibit seepage faces. We focus on a representative case of the steady state flows governed by a nonlinear Darcy–Buckingham law with physical constraints on subsurface-atmosphere boundaries. This leads to the formulation of the problem as a variational inequality. The solutions to this problem are investigated using an adaptive finite element method based on a dual-weighted a posteriori error estimate, derived with the aim of reducing error in a specific target quantity. The quantity of interest is chosen as volumetric water flux across the seepage face, and therefore depends on an a priori unknown free boundary. We apply our method to challenging numerical examples as well as specific case studies, from which this research originates, illustrating the major difficulties that arise in practical situations. We summarise extensive numerical results that clearly demonstrate the designed method produces rapid error reduction measured against the number of degrees of freedom.

scholarly journals An adaptive finite element DtN method for the elastic wave scattering by biperiodic structures

2021 ◽  
Author(s):  
Gang Bao ◽  
Xue Jiang ◽  
Peijun Li ◽  
Xiaokai Yuan
Keyword(s):  
Finite Element ◽  
Error Estimate ◽  
Elastic Wave ◽  
Posteriori Error ◽  
A Posteriori Error Estimate ◽  
Posteriori Error Estimate ◽  
Adaptive Finite Element ◽  
A Posteriori ◽  
Truncation Parameter ◽  
A Posteriori Error

Consider the scattering of a time-harmonic elastic plane wave by a bi-periodic rigid surface. The displacement of elastic wave motion is modeled by the three-dimensional Navier equation in an unbounded domain above the surface. Based on the Dirichlet-to-Neumann (DtN) operator, which is given as an infinite series, an exact transparent boundary condition is introduced and the scattering problem is formulated equivalently into a boundary value problem in a bounded domain. An a posteriori error estimate based adaptive finite element DtN method is proposed to solve the discrete variational problem where the DtN operator is truncated into a finite number of terms. The a posteriori error estimate takes account of the finite element approximation error and the truncation error of the DtN operator which is shown to decay exponentially with respect to the truncation parameter. Numerical experiments are presented to illustrate the effectiveness of the proposed method.

scholarly journals Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation

2012 ◽  
Vol 11 (1) ◽  
pp. 179-214 ◽  
Author(s):  
M. Holst ◽  
J.A. McCammon ◽  
Z. Yu ◽  
Y.C. Zhou ◽  
Y. Zhu
Keyword(s):  
Finite Element ◽  
Boltzmann Equation ◽  
Geometric Model ◽  
A Priori ◽  
Molecular Surface ◽  
Adaptive Finite Element Method ◽  
Adaptive Finite Element ◽  
Regularization Technique ◽  
Poisson Boltzmann ◽  
Poisson Boltzmann Equation

AbstractWe consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the two-term regularization technique for the continuous problem recently proposed by Chen, Holst and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation, the first provably convergent discretization and also allowed for the development of a provably convergent AFEM. However, in practical implementation, this two-term regularization exhibits numerical instability. Therefore, we examine a variation of this regularization technique which can be shown to be less susceptible to such instability. We establish a priori estimates and other basic results for the continuous regularized problem, as well as for Galerkin finite element approximations. We show that the new approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme for the new regularized problem and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is one of the first results of this type for nonlinear elliptic problems, is based on using continuous and discrete a prioriL∞ estimates. To provide a high-quality geometric model as input to the AFEM algorithm, we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures, based on the intrinsic local structure tensor of the molecular surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages of the new regularization scheme are demonstrated with FETK through comparisons with the original regularization approach for a model problem. The convergence and accuracy of the overall AFEMalgorithmis also illustrated by numerical approximation of electrostatic solvation energy for an insulin protein.

ADAPTIVE FINITE ELEMENT METHODS FOR REACTIVE COMPRESSIBLE FLOW

1999 ◽  
Vol 09 (02) ◽  
pp. 211-241 ◽  
Author(s):  
ROBERT SANDBOGE
Keyword(s):  
Finite Element ◽  
Compressible Flow ◽  
Error Control ◽  
Posteriori Error ◽  
Posteriori Error Estimate ◽  
Stability Factor ◽  
Adaptive Finite Element ◽  
Adaptive Finite Element Methods ◽  
Adaptive Error Control ◽  
The Stability

We extend the adaptive streamline diffusion finite element method for compressible flow in conservation variables using P1× P0 space–time elements to include chemical reactions. The adaptive error control is based on an a posteriori error estimate involving a stability factor, which is estimated numerically. We prove for a model problem that the stability factor is bounded by a moderate constant.

scholarly journals An Adaptive Finite Element PML Method for the Acoustic-Elastic Interaction in Three Dimensions

2017 ◽  
Vol 22 (5) ◽  
pp. 1486-1507 ◽  
Author(s):  
Xue Jiang ◽  
Peijun Li
Keyword(s):  
Finite Element ◽  
Scattering Problem ◽  
Elastic Solid ◽  
Elastic Interaction ◽  
Posteriori Error ◽  
Three Dimensions ◽  
Adaptive Finite Element Method ◽  
Posteriori Error Estimate ◽  
Computational Domain ◽  
Adaptive Finite Element

AbstractConsider the scattering of a time-harmonic acoustic incident wave by a bounded, penetrable, and isotropic elastic solid, which is immersed in a homogeneous compressible air or fluid. The paper concerns the numerical solution for such an acoustic-elastic interaction problem in three dimensions. An exact transparent boundary condition (TBC) is developed to reduce the problem equivalently into a boundary value problem in a bounded domain. The perfectly matched layer (PML) technique is adopted to truncate the unbounded physical domain into a bounded computational domain. The well-posedness and exponential convergence of the solution are established for the truncated PML problem by using a PML equivalent TBC. An a posteriori error estimate based adaptive finite element method is developed to solve the scattering problem. Numerical experiments are included to demonstrate the competitive behavior of the proposed method.

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