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A Galerkin formulation of the MIB method for three dimensional elliptic interface problems.

ABSTRACT: We develop a three dimensional (3D) Galerkin formulation of the matched interface and boundary (MIB) method for solving elliptic partial differential equations (PDEs) with discontinuous coefficients, i.e., the elliptic interface problem. The present approach builds up two sets of elements respectively on two extended subdomains which both include the interface. As a result, two sets of elements overlap each other near the interface. Fictitious solutions are defined on the overlapping part of the elements, so that the differentiation operations of the original PDEs can be discretized as if there was no interface. The extra coefficients of polynomial basis functions, which furnish the overlapping elements and solve the fictitious solutions, are determined by interface jump conditions. Consequently, the interface jump conditions are rigorously enforced on the interface. The present method utilizes Cartesian meshes to avoid the mesh generation in conventional finite element methods (FEMs). We implement the proposed MIB Galerkin method with three different elements, namely, rectangular prism element, five-tetrahedron element and six-tetrahedron element, which tile the Cartesian mesh without introducing any new node. The accuracy, stability and robustness of the proposed 3D MIB Galerkin are extensively validated over three types of elliptic interface problems. In the first type, interfaces are analytically defined by level set functions. These interfaces are relatively simple but admit geometric singularities. In the second type, interfaces are defined by protein surfaces, which are truly arbitrarily complex. The last type of interfaces originates from multiprotein complexes, such as molecular motors. Near second order accuracy has been confirmed for all of these problems. To our knowledge, it is the first time for an FEM to show a near second order convergence in solving the Poisson equation with realistic protein surfaces. Additionally, the present work offers the first known near second order accurate method for C1 continuous or H2 continuous solutions associated with a Lipschitz continuous interface in a 3D setting.

SUBMITTER: Xia K

PROVIDER: S-EPMC4190480 | biostudies-literature | 2014 Oct

REPOSITORIES: biostudies-literature

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A Galerkin formulation of the MIB method for three dimensional elliptic interface problems.

Xia Kelin K Wei Guo-Wei GW

Computers & mathematics with applications (Oxford, England : 1987) 20141001 7

We develop a three dimensional (3D) Galerkin formulation of the matched interface and boundary (MIB) method for solving elliptic partial differential equations (PDEs) with discontinuous coefficients, i.e., the elliptic interface problem. The present approach builds up two sets of elements respectively on two extended subdomains which both include the interface. As a result, two sets of elements overlap each other near the interface. Fictitious solutions are defined on the overlapping part of the ...[more]

PMID: 25309038

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Project description:PurposeTo present a novel non-parametric algorithm for detecting the position of the human eye limbus in three dimensions and a new dynamic method for measuring the full 360° visible iris boundary known as white-to-white distance along the eye horizontal line.MethodsThe study included 88 participants aged 23 to 65 years (37.7±9.7), 47 females and 41 males. Clinical characteristics, height data and the apex coordinates and 1024×1280 pixel digital images of the eyes were taken by an Eye Surface Profiler and processed by custom-built MATLAB codes. A dynamic light intensity frequency based white-to-white detection process and a novel three-dimensional method for limbus detection is presented.ResultsEvidence of significant differences (p<0.001) between nasal-temporal and superior-inferior white-to-white distances in both right and left eyes were found (nasal-temporal direction; 11.74±0.42 mm in right eyes and 11.82±0.47 mm in left eyes & superior-inferior direction; 11.52±0.45 mm in right eyes and 11.55±0.46 mm in left eyes). Average limbus nasal-temporal diameters were 13.64±0.55 mm for right eyes, and 13.74±0.40 mm for left eyes, however the superior-inferior diameters were 13.65±0.54 mm, 13.75±0.38 mm for right and left eyes, respectively. No significant difference in limbus contours has been observed either between the nasal-temporal direction (p = 0.91) and the superior-inferior direction (p = 0.83) or between the right (p = 0.18) and left eyes (p = 0.16). Evidence of tilt towards the nasal-temporal side in the three-dimensional shape of the limbus was found. The right eyes mean limbus contour tilt around the X-axis was -0.3±1.35° however, their mean limbus contour tilt around the Y-axis was 1.76±0.9°. Likewise, the left eyes mean limbus contour tilt around the X-axis was 0.77±1.25° and the mean limbus contour tilt around the Y-axis was -1.54±0.89°.ConclusionsThe white-to-white distance in the human eye is significantly larger in the nasal-temporal direction than in the superior-inferior direction. The human limbus diameter was found not to vary significantly in these directions. The 3D measures show that the limbus contour does not lay in one plane and tends to be higher on the nasal-inferior side of the eye.

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A Galerkin formulation of the MIB method for three dimensional elliptic interface problems.

Publications

A Galerkin formulation of the MIB method for three dimensional elliptic interface problems.

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