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# A Nonlocal Operator Method for Partial Differential Equations with Application to Electromagnetic Waveguide Problem

Timon Rabczuk1,2,*, Huilong Ren3, Xiaoying Zhuang4,5

Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam.
Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Viet Nam.
Institute of Structural Mechanics, Bauhaus -University Weimar, 99423 Weimar, Germany.
Institute of Continuum Mechanics, Leibniz University Hannover, Hannover, Germany.
State Key Laboratory of Disaster Reduction in Civil Engineering, College of Civil Engineering, Tongji University, Shanghai, 200092, China.

* Corresponding Author: Timon Rabczuk. Email: .

Computers, Materials & Continua 2019, 59(1), 31-55. https://doi.org/10.32604/cmc.2019.04567

## Abstract

A novel nonlocal operator theory based on the variational principle is proposed for the solution of partial differential equations. Common differential operators as well as the variational forms are defined within the context of nonlocal operators. The present nonlocal formulation allows the assembling of the tangent stiffness matrix with ease and simplicity, which is necessary for the eigenvalue analysis such as the waveguide problem. The present formulation is applied to solve the differential electromagnetic vector wave equations based on electric fields. The governing equations are converted into nonlocal integral form. An hourglass energy functional is introduced for the elimination of zero-energy modes. Finally, the proposed method is validated by testing three classical benchmark problems.

## Keywords

APA Style
Rabczuk, T., Ren, H., Zhuang, X. (2019). A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem. Computers, Materials & Continua, 59(1), 31-55. https://doi.org/10.32604/cmc.2019.04567
Vancouver Style
Rabczuk T, Ren H, Zhuang X. A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem. Comput Mater Contin. 2019;59(1):31-55 https://doi.org/10.32604/cmc.2019.04567
IEEE Style
T. Rabczuk, H. Ren, and X. Zhuang "A Nonlocal Operator Method for Partial Differential Equations with Application to Electromagnetic Waveguide Problem," Comput. Mater. Contin., vol. 59, no. 1, pp. 31-55. 2019. https://doi.org/10.32604/cmc.2019.04567

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