Local support

Definition

Local support refers to the property of certain basis functions in finite element methods where they are non-zero only within a limited region of the domain. This characteristic allows for efficient computations and simplifies the assembly of the global stiffness matrix, as only nearby elements contribute to the calculations. By focusing on local interactions, local support helps reduce the computational cost and enhances the accuracy of numerical simulations.

5 Must Know Facts For Your Next Test

1. Local support is crucial for finite element analysis as it minimizes the number of interactions needed during calculations.
2. The use of local support leads to sparse matrices, which are easier to manipulate and store compared to dense matrices.
3. Different types of finite element methods may use various basis functions, each exhibiting local support tailored for specific applications.
4. In practical applications, local support allows for localized refinement of meshes, leading to improved accuracy without drastically increasing computational resources.
5. Understanding local support is essential for implementing adaptive meshing techniques that dynamically adjust element sizes based on solution gradients.

Review Questions

• How does local support influence the efficiency of finite element methods?
  • Local support significantly enhances the efficiency of finite element methods by limiting the interaction between elements to only those in close proximity. This property reduces the computational workload since it avoids unnecessary calculations involving distant elements. As a result, this leads to faster assembly of matrices and lower memory usage, making simulations more manageable and efficient.
• In what ways do basis functions with local support improve the accuracy of numerical simulations in finite element methods?
  • Basis functions with local support allow for better approximation of solutions in finite element methods by capturing localized behaviors and variations within the domain. Since these functions are non-zero only within a limited area, they provide precise control over how closely the solution can follow complex geometries or boundary conditions. This localized approach ultimately leads to higher accuracy in representing the physical problem being modeled.
• Evaluate the role of local support in adaptive meshing techniques used in finite element methods and its impact on computational resources.
  • Local support plays a pivotal role in adaptive meshing techniques by allowing for targeted refinement in areas where higher resolution is needed while maintaining coarser meshes elsewhere. This strategic approach minimizes computational resources since it limits the number of elements needing fine discretization. By focusing computational power on critical regions, local support ensures that simulations remain efficient while achieving necessary accuracy, significantly impacting overall performance in practical applications.

Related terms

Basis Functions: Functions used in finite element methods to approximate the solution over an element, characterized by their local support properties.

Element Connectivity: The arrangement of nodes and elements in a finite element mesh that defines how elements are connected and interact with each other.

Global Stiffness Matrix: A matrix that represents the overall stiffness of the entire finite element model, constructed by assembling contributions from individual elements.